Dissertation - Nominalism in Mathematics: Modality and Naturalism

NOMINALISM IN MATHEMATICS: MODALITY AND NATURALISM

by

JAMES S.J. SCHWARTZ

DISSERTATION

Submitted to the Graduate School

of Wayne State University,

Detroit, Michigan

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

2013

MAJOR: PHILOSOPHY

Approved by:

Advisor Date

c©COPYRIGHT BY

JAMES S.J. SCHWARTZ

2013

All Rights Reserved

DEDICATION

For my mother

ii

ACKNOWLEDGEMENTS

Special thanks are due to the chair of my dissertation committee, Susan Vineberg,

who has carefully reviewed far too many drafts of each chapter. Without her guidance

this project would not have been possible. Many thanks are of course due to the other

members of my committee: Eric Hiddleston, Michael McKinsey, and Robert Bruner. I

am also indebted to Sean Stidd (my unofficial “fifth committee member”), John Halpin,

and Matt McKeon for discussion along the way, as well as to various other philosophers,

colleagues, and friends: David Baxter, Sharon Berry, Daniel Blaser, Marcus Cooper, John

Corvino, Michael Ernst, Travis Figg, David Garfinkle, Herbert Granger, Geoffrey Hellman,

Timothy Kirschenheiter, Teresa Kouri, Øystein Linnebo, Lawrence Lombard, Penelope

Maddy, Gonzalo Munevar, Gregory Novack, Christopher Pincock, Lawrence Powers,

Daniel Propson, Mark Rigstad, Bruce Russell, Scott Shalkowski, Stewart Shapiro, and

Daniel Yeakel.

An abridged version of the second chapter was read at a Philosophy Department

Colloquium at Wayne State University in April of 2012 and at the 2012 Meeting of the

Society for Exact Philosophy in October of 2012. An earlier version of the fourth chapter

was the topic of a Natural Philosophy Workshop at Oakland University in March of

2012, and an abridged version of this chapter was presented at the 4th Annual Graduate

Exhibition at Wayne State University in March of 2013. I thank those in attendance at the

above events for comments and discussion.

Chapter three has improved greatly from extensive comments from Eric Hiddleston

and from Michael McKinsey. The chapters on naturalism (four and five) are the result

of many rounds of comments from Susan Vineberg. Their influence on these sections of

the dissertation is warmly acknowledged; I fear they do not receive enough credit in the

footnotes—especially for helping me to think more clearly about which ideas were worth

developing and which were not.

Finally, I would to thank my mother, Cynthia Johnson, whose encouragement and sup-

iii

port throughout my entire education has made possible my completion of this dissertation.

iv

TABLE OF CONTENTS

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

II The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

II.1 What is the Philosophy of Mathematics? . . . . . . . . . . . . . . . . . 2

II.2 What is Nominalism? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

III Modality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

III.1 Modality in Philosophy of Mathematics . . . . . . . . . . . . . . . . . 6

III.2 Shapiro’s Challenge to Nominalism . . . . . . . . . . . . . . . . . . . 8

III.3 Modality and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 14

IV Naturalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

IV.1 Reflections on Burgess . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

IV.2 Reflections on Maddy . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

V Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Part 1: Modality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1 Modality in the Philosophy of Mathematics . . . . . . . . . . . . . . . . . . . . . 28

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2 Chihara’s Constructibility Theory . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.2.1 The Constructibility Theory . . . . . . . . . . . . . . . . . . . . . . . . 30

1.2.2 A Closer Look at Modality in Constructibility Theory . . . . . . . . . 34

1.3 Hellman’s Modal Structuralism . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.3.1 A Closer Look at Modality in Modal Structuralism . . . . . . . . . . . 47

1.4 Field’s Fictionalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

v

1.4.1 A Closer Look at Modality in Field’s Fictionalism . . . . . . . . . . . 54

1.4.2 Field and Justifying Modal Assertions . . . . . . . . . . . . . . . . . . 58

1.5 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

1.5.1 Balaguer’s Fictionalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

1.5.2 Leng’s Fictionalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2 Shapiro’s Challenge to the use of Modality in Nominalist Theories . . . . . . . 71

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.2 Modality and Ontology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.2.1 The Emperor’s New Epistemology . . . . . . . . . . . . . . . . . . . . 75

2.2.2 The Emperor’s New Ontology . . . . . . . . . . . . . . . . . . . . . . 83

2.3 First Reply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.4 The Paraphrase Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.4.1 Nominalist Paraphrase as Synonymy . . . . . . . . . . . . . . . . . . 95

2.5 Reply to the Paraphrase Response . . . . . . . . . . . . . . . . . . . . . . . . 100

2.6 The Structuralist Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

2.6.1 Resnik and Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.6.2 Shapiro and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

2.7 Reply to the Structuralist Response . . . . . . . . . . . . . . . . . . . . . . . . 125

2.7.1 Withering Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

2.8 Shapiro’s Challenge: What Exactly is the Problem? . . . . . . . . . . . . . . . 137

3 Reducing Modality as a Solution to Shapiro’s Challenge . . . . . . . . . . . . . 146

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.2 Reduction and Shapiro’s Challenge . . . . . . . . . . . . . . . . . . . . . . . . 149

3.2.1 Mission Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

3.3 Justifying Modal Assertions Under Lewis’s Reduction . . . . . . . . . . . . . 156

3.3.1 Lewis’s Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

vi

3.3.2 Lewis and Justifying Modal Assertions . . . . . . . . . . . . . . . . . 158

3.4 Possible Worlds and Functional Roles . . . . . . . . . . . . . . . . . . . . . . 164

3.5 Lessons for Shapiro and Set Theory . . . . . . . . . . . . . . . . . . . . . . . . 166

3.5.1 The Set-Theoretic Reduction . . . . . . . . . . . . . . . . . . . . . . . . 166

3.5.2 Justifying Modal Assertions Under the Set-Theoretic Reduction . . . 169

3.6 Living With Primitive Modality . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Part 2: Naturalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4 Reflections on Burgess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

4.2 The Master Argument Against Nominalism . . . . . . . . . . . . . . . . . . . 185

4.2.1 Burgess’s Naturalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

4.2.2 The Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.3 Replies in the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.3.1 The Scientific Merits of Nominalistic Reinterpretation . . . . . . . . . 201

4.3.2 Scientific Merits Redux: Chihara and the Attitude-Hermeneuticist . . 206

4.3.3 The False Dilemma Reply . . . . . . . . . . . . . . . . . . . . . . . . . 208

4.4 My Reply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

4.4.1 The Tonsorial Question . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

4.5 Why Burgess is not a (Moderate) Platonist . . . . . . . . . . . . . . . . . . . . 227

5 Reflections on Maddy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.2 Second Philosophy of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . 239

5.2.1 Two Provisional Second-Philosophical Objections to Modal Nomi-

nalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

5.3 The Method-Affirming Objection to Modal Nominalism . . . . . . . . . . . . 249

5.3.1 A Miscellany of Objections . . . . . . . . . . . . . . . . . . . . . . . . . 250

vii

5.3.2 Method-Affirming as a Prophylactic . . . . . . . . . . . . . . . . . . . 254

5.3.3 Method- and Result-Rejecting: The Good and The Bad . . . . . . . . 257

5.3.4 The Method-Affirming Objection Reconsidered . . . . . . . . . . . . . 258

5.3.5 The Method-Affirming Objection: Coda . . . . . . . . . . . . . . . . . 265

5.4 The Method-Contained Objection to Modal Nominalism . . . . . . . . . . . 266

5.4.1 Thin Realism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

5.4.2 Arealism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

5.4.3 There is no Difference Here . . . . . . . . . . . . . . . . . . . . . . . . 274

5.4.4 The Method-Contained Objection: Coda . . . . . . . . . . . . . . . . . 277

5.5 How Much Naturalism is Too Much Naturalism? . . . . . . . . . . . . . . . . 279

5.6 Conclusion: Naturalism and Modal Nominalism . . . . . . . . . . . . . . . . 283

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

Autobiographical Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

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1

Introduction

I Thesis

This dissertation is a defense of certain modal nominalist strategies in the philosophy of

mathematics against two varieties of criticism. My primary thesis, which is defended in

Part I (chapters one, two, and three), is that modal nominalists do not encounter uniquely

challenging difficulties in regards to justifying the modal assertions that figure in modal

nominalist theories of mathematics. My secondary thesis, which is defended in Part II

(chapters four and five), is that modal nominalism does not engender any serious conflict

with the practice of mathematics, and thus is consistent with the naturalistic impulse to

respect science and mathematics. In defending these theses I hope to establish modal

nominalism as a viable and attractive philosophy of mathematics, one that can provide

satisfactory answers to many of philosophy of mathematics’ longstanding questions.

My goal in this introduction is to explain the provenance of these theses by examining

the following questions: What is the philosophy of mathematics? What is nominalism?

What is modal nominalism? What has modality got to do with nominalism in mathematics,

and what, if anything, should a modal nominalist find troubling about this? What is

naturalism, and why should anyone suppose that naturalism and modal nominalism are

incompatible?

James S.J. Schwartz Nominalism in Mathematics

Introduction 2 The Basics

II The Basics

II.1 What is the Philosophy of Mathematics?

The philosophy of mathematics is a broad-ranging subdiscipline of analytic philosophy

tasked with answering questions such as: “What is mathematics about?” “What is the best

characterization of what the practicing mathematician does?” “What is the methodology of

mathematics, be it pure or applied?” “How do mathematicians decide whether to adopt

new theories or axioms?” “Do there exist mathematical objects?” “What kind of a logical

foundation is necessary for doing mathematics?” “Why is mathematical reasoning so

successful in scientific applications?”. . . and the list could go on. Nevertheless, the picture

that emerges is one according to which philosophers of mathematics are interested in the

nature of mathematics as well as the methodology of its practice.

With few exceptions, mathematicians have rather little to say about the nature of their

discipline; they would appear to be interested more in activities like seeking out fruitful

mathematical concepts and axioms, constructing proofs, and investigating the structural

properties of formal systems, than in the activity of determining whether the axioms

utilized in their proofs are made true by any particular objects. Questions such as “What

can be proven from a particular set of mathematical axioms?” and “What properties are

shared by all dense linear orderings?” do not strike me as overtly philosophical questions,

at least when they are asked from the standpoint of someone who is trying to understand

mathematics as it is practiced. Related to the case of proof construction, perhaps there are

good philosophical reasons—although I doubt this—for rejecting classical logic. But to

argue on philosophical grounds that mathematicians have no reason to believe in the

conclusions of nonconstructive proofs seems not so much an exercise in attempting to

understand mathematics as it is a case of attempting to tell mathematicians what they can

and cannot do.

On the other hand, questions as to whether a set of mathematical axioms holds true of


Introduction 3 What is Nominalism?

anything, and moreover, whether there is any uniquely mathematical ontology, do strike me

as philosophical questions. And provided that the philosopher does not say anything to

sully the practice of mathematics, she should not be impeded in her attempts to understand

what exactly the truth of a mathematical theory comes to, if indeed mathematical theories

are the kinds of things she ought to regard as true.

In this dissertation I will principally be concerned with the question about whether

mathematical objects exist. I aim to defend several theories, viz., Charles Chihara’s Con-

structibility Theory, Geoffrey Hellman’s Modal Structuralism, and Hartry Field’s fiction-

alism, each of which purports to offer accounts of, e.g., mathematical knowledge, and

the content of mathematical claims, in ways that do not presuppose or otherwise require

the existence of mathematical objects. Views such as these are generally described as

nominalistic accounts of mathematics. But what does it mean for an account of mathematics

to be nominalistic? And are Constructibility Theory, Modal Structuralism, and fictionalism

genuinely nominalistic?

II.2 What is Nominalism?

Nominalism is a philosophical thesis according to which abstract objects, such as universals,

do not exist. It is often contrasted with a thesis known sometimes as ‘realism’ and other

times as ‘platonism,’1 which holds that there do exist abstract objects of various kinds.

Nominalism in mathematics is the philosophical thesis that there do not exist any abstract

mathematical objects, such as numbers, functions, sets, and so forth. It is often contrasted

with platonism in mathematics, which holds that there do exist abstract mathematical

objects. With the exception of the remainder of this section, I will drop the lengthy

phrases ‘nominalism in mathematics’ and ‘platonism in mathematics,’ using just the terms

‘nominalism’ and ‘platonism’ to refer to their respective positions in the philosophy of

mathematics. On the (rare!) occasions when I discuss the general views, I will use terms

1Note the lower-case ‘p’ to distance contemporary platonism from genuine Platonic doctrines. I shall usethe terms ‘realism’ and ‘platonism’ interchangeably throughout this dissertation.



such as ‘general nominalism/platonism’ and ‘nominalism/platonism in general.’

The distinction between nominalism/realism in general and nominalism/realism in

mathematics is important. A defender of nominalism (or realism) in mathematics is not, on

pain of contradiction, obliged to provide concomitant defenses of nominalism (or realism)

in other areas. There is conceptual space available for a person interested in denying the

existence of abstract mathematical objects, while accepting the existence of universals or

other kinds of abstracta. Similarly, one might be a realist about mathematical objects, while

denying that universals and other kinds of abstracta exist. Perhaps there is some peculiar

kind of inconsistency at play in such views, but whatever inconsistency may be present, I

can make no sense of saying that it is a logical inconsistency.

The nominalist faces a number of challenges within the philosophy of mathematics.

Following W.V. Quine and Hilary Putnam, many have argued on naturalistic grounds

that abstract mathematical objects are somehow indispensable for describing the world in

a scientifically perspicuous way. The best scientific theories include mathematics; one

is compelled to believe the best scientific theories—but how could one do this without

believing the constituent mathematical assertions of the best scientific theories? Or, to put

the point more strongly, it seems rational to regard the best scientific theories as being true,

but how could such theories be regarded as true unless their constituent mathematical

assertions are similarly regarded as true? Prima facie, mathematical assertions are about

things like numbers, functions, vectors, and spaces—how could such assertions be true

without there existing numbers, functions, vectors, and spaces?

Much of the work done by nominalists in the 1980s consisted of attempts to rebut the

indispensability arguments. Various strategies were proposed, ranging from denying that

the best scientific theories require mathematics, to denying that mathematical assertions

are best understood as being about abstract mathematical objects. A great deal of this

work, viz., the work stemming from Field’s fictionalist program, has been motivated by

epistemological concerns. In the previous decade the causal theory of knowledge had



won over many philosophers. If one must be causally connected to an object in order to

have knowledge about it, and if abstract mathematical objects are acausal, then no one is

causally connected to any mathematical objects, from which it follows that no one can know

that they exist, making it a great mystery as to how mathematicians (and ordinary folks)

apparently know so many things about them. Although the causal theory of knowledge

is currently out of favor, many believe there are lingering problems about how best to

explain why it is that mathematicians come to possess their very reliable mathematical

beliefs. What mechanisms ensure that mathematical reasoning, which is done largely

without reference to metaphysical hypotheses about the nature of mathematical truth,

accesses truths about mathematical objects, platonistically construed? What explanation

can be given for the correlation between statements such as “complex numbers exist” and

“mathematicians believe that complex numbers exist?” For, according to the traditional

platonist picture, mathematical objects in no way interact with human beings. This suggests

that mathematicians would believe that complex numbers existed even if the facts about

the mathematical realm were entirely different, and serves to undermine any claim that

the mathematician thereby knows that complex numbers exist.

Nominalism in mathematics, at least in the primary sense, requires the outright denial

that mathematical objects exist. That is the positive component of nominalism in mathe-

matics, at any rate. But notice that the epistemological concerns just mentioned function

primarily as objections to platonism in mathematics, i.e., as criticisms of the claim that

anyone has good reasons for supposing that mathematical objects exist. This identifies a

further, negative component of nominalism in mathematics—to undercut platonist argu-

ments that, if sound, would establish the existence of mathematical objects. As it turns

out, most of the theories of mathematics that have been described as “nominalist” theories

of mathematics, including those defended in this dissertation, do not seek to decisively

establish the non-existence of mathematical objects, but are instead constructed for the

purpose of undermining platonism in mathematics. Thus, most “nominalist” theories


Introduction 6 Modality

of mathematics are not nominalist in the primary sense but are instead nominalist in a

weaker, secondary sense, in which their truth is compatible with (but does not require) the

non-existence of mathematical objects.

Though I welcome efforts to establish the truth of nominalism (in mathematics and

in general) in the primary sense, this dissertation is, strictly speaking, a defense of sev-

eral theories of mathematics that are “nominalist” in the secondary sense. That is, this

dissertation is a defense of several theories—Chihara’s Constructibility Theory, Hellman’s

Modal Structuralism, and Field’s fictionalism—that were constructed for the purpose of

showing that it is not necessary to assume the existence of mathematical objects in order to

provide an account of the content of mathematical claims. In particular, I aim to defend

the specific metaphysical claims that these “nominalists” make in the construction and

advancement of their theories of mathematics. In this dissertation I will often use the

term ‘modal nominalism’ to refer to these views as a group (which are loosely related

in that each invokes modality in important ways). Thus, when I discuss, defend, and

offer arguments in support of modal nominalism, I am not trying to establish the truth of

nominalism in mathematics in the primary sense; I am only trying to establish nominalism

in mathematics in the weaker, secondary sense.

III Modality

What distinguishes modal nominalist theories in philosophy of mathematics is their

use of modality in eschewing commitment to mathematical objects. But why do modal

nominalists opt to embrace modality in order to eschew commitment to mathematical

objects? And how can modality be used to facilitate this task?

III.1 Modality in Philosophy of Mathematics

As a first example, consider the Constructibility Theory of (Chihara 1990). Chihara avoids

postulating the existence of mathematical objects by replacing mathematical existence

assertions with assertions about the constructibility of open-sentence tokens, through


Introduction 7 Modality in Philosophy of Mathematics

which he hopes to capture a simple type-theoretic framework. These constructibility asser-

tions are taken to express statements about the metaphysical possibility of constructing

open-sentence tokens. A second example is Hellman’s Modal Structuralism (1989). On

Hellman’s view, one can construe mathematical assertions as assertions about what holds

within various possible mathematical structures, avoiding the need to quantify over any ac-

tually existing mathematical objects. Hellman takes such assertions to be statements about

what is primitively logically possible. A final example is Field’s fictionalism (1980). Field’s

account of the scientific applications of mathematics holds that mathematics is conservative in

the sense that any nominalistic consequence of a nominalized physical theory that includes

mathematics is a consequence of the nominalized physical theory alone. In articulating

this view he develops an account of mathematical practice which holds that mathematical

knowledge is just logical knowledge. The mathematician can be described simply as

person who investigates what follows from mathematical assumptions. According to Field,

accounting for this kind of logical exercise requires premises no stronger than those which

assert the consistency of the relevant mathematical assumptions, which, for Field, comes

to the assertion of the primitive logical possibility of their conjunction.

More details on these three projects are given in the first chapter, but it should now be

clear that the use of modality performs important work in modal nominalist philosophies

of mathematics. Platonists have seen fit to criticize them on this score. If modality

must be invoked in order to eschew commitment to mathematical objects, this would

appear to improve this situation only if modality does not raise epistemological and

metaphysical difficulties that are just as serious as those surrounding the mathematical

objects under exile. For instance, it might be that the best story about the truth-conditions

for modal assertions involves some postliminary increase in ontology. Proposed reductive

bases for modality have included full-blooded possible worlds, maximally consistent sets

of propositions, and maximal combinations of states of affairs, each of which have at least

a tinge of platonism to them. If the modal nominalist opts for such reductions, then two


Introduction 8 Shapiro’s Challenge to Nominalism

apparently undesirable consequences follow. First is that each proposed reduction requires

an increase in ontology (a move that is clearly problematic for those interested in pursuing

a general nominalism). Second is that modal nominalists must further secure a means

for justifying assertions about the proposed reductive bases. The modal nominalist could

hardly claim to have improved on the platonist’s account of mathematical knowledge if she

proposes only to exchange one set of difficulties (justifying existence and knowledge claims

about abstract mathematical objects) for another set of difficulties (justifying existence and

knowledge claims about possible worlds, sets of propositions, etc.). Modal nominalists

believe that they can avoid any such increase in ontology by maintaining that possibility

and necessity are primitive and unanalyzable notions, but doing so only serves to raise

questions about how it is possible to justify assertions about what is primitively necessary

and possible (e.g., to explain why it is the case that the axioms of Zermelo-Fraenkel set

theory are jointly possible and to justify such an assertion on nominalistically acceptable

grounds).

III.2 Shapiro’s Challenge to Nominalism

The criticism that modal nominalism raises serious questions concerning the modal nom-

inalist’s ability to justify modal assertions is voiced most forcefully by Stewart Shapiro

(1997). Any interpretation of mathematics that is powerful enough to capture ordinary

mathematical reasoning must provide a scheme for translating classical mathematical

assertions into nominalized assertions.2 Shapiro’s insight is in recognizing that these trans-

lations can also be undone. That is, the modal nominalist’s translations can be translated

back into their original platonistic counterparts. This is a significant result for Shapiro,

who, as a mathematical structuralist, views mathematics as the study of the relations that

hold between objects (e.g., number-theoretic relations) rather than an investigation of the

objects themselves (e.g., numbers). The intertranslatability between ordinary mathematical

2E.g., in Hellman’s case, the assertion that ‘Peano Arithmetic implies A’ can be “nominalized” byreconstruing it as ‘Necessarily, if X satisfies the Peano axioms then X implies A.’



assertions and the modal nominalist’s reconstructions suffices to demonstrate that both

accounts capture the same mathematical structure. According to Shapiro, mathematical struc-

tures are freestanding objects, akin to ante rem universals, and any view that characterizes

a mathematical structure is thereby committed to that structure’s existence. Thus, modal

nominalist views are ultimately committed to abstract objects in the form of mathematical

structures. Therefore, the modal nominalist is unsuccessful in her attempts to eschew

commitment to mathematical entities.

To add injury to insult, Shapiro draws attention to the fact that modal nominalists do

not have available any nominalistically acceptable reductive bases for the modal notions

they employ in their theories. It is commonly thought that the logical modalities—the kind

of modality used by Hellman and Field—can be reduced model-theoretically.3 Although

there is less agreement in the case of the metaphysical modalities—the kind of modality

used by Chihara—some believe that these can be reduced by countenancing either possible

worlds or one among a sundry of possible worlds surrogates. The possibility of reducing

the modal notions to non-modal notions is significant in the following sense: A nominalist

such as Hellman appears stuck with, e.g., the logical possibility of the existence of a model

of Peano Arithmetic as a brute and unanalyzable fact—an assertion the truth of which is

not amenable to explanation. However, if the logical modalities can be reduced to facts

about models, then one can explain the logical possibility of p by pointing to the relevant

non-modal facts about some model or other. Thus, the logical possibility of the existence

of a model of Peano Arithmetic can be demonstrated by constructing a model satisfying

the axioms of Peano Arithmetic. If the metaphysical modalities can be reduced, then one

can give an explanation for why p is metaphysically possible by pointing to the relevant

non-modal facts about some possible world (or possible world surrogate). Reduction

thereby increases one’s explanatory resources and would appear to free one from needing

to devise a special epistemology for modality (because modal knowledge just is knowledge

3Thus the association with logical necessity as truth-in-all-models and logical possibility as truth-in-a-model.



about the reductive base).

According to Shapiro, the modal nominalists in question seem happy to accept prim-

itive modal notions, thereby incurring the challenge of justifying the various primitive

modal assertions that figure in their accounts of mathematics. Shapiro also insists that

modal nominalists (overtly or covertly) rely on the platonistic, model-theoretic account

of the logical modalities, and consequently, that they cannot coherently engage in modal

reasoning in isolation from reasoning in model-theoretic systems of modal logic. Nomi-

nalists such as Chihara and Field have offered responses to the latter complaint, but none

have done anything to rebut the justificatory criticism identified in the former complaint.

Modal nominalists thereby do appear to be burdened with intractable problems involving

modality. My overall aim in Part I of this dissertation (chapters one, two, and three) is to

show that modal nominalists are not unique in carrying this justificatory burden.

There is ordinarily thought to be a tradeoff between ontology and ideology. One can

deflate one’s ontology by bloating one’s ideology, and vice versa. In appealing to the

modal notions to eschew commitment to mathematical objects, the modal nominalist

accepts what appears to her to be a bargain; for the price of a small increase in ideology

she can vastly simplify her ontology. But if Shapiro is right, matters are not nearly so

simple. In the first place, the modal nominalist’s reconstructions do not reduce her

ontology—she is unwittingly committed to mathematical structures. In the second place,

the theoretical resources she requires in order to produce her reconstructions inflate her

ideology and burden her with a new challenge—that of explaining and justifying, in a

non-platonistic way, why it is the case that various mathematical theories are possible (or

that it is possible to construct open-sentence tokens of various kinds). In comparison with

platonism (burdened only by its vast ontology), modal nominalism is the clear runner-up.

My response to Shapiro addresses both the criticism that modal nominalist reconstruc-

tions of mathematics do not reduce ontology, and also the accusation that the modal

nominalist incurs a novel burden (one not shared with the platonist) concerning the



justification of modal assertions. After providing a detailed reconstruction of Shapiro’s

arguments, I start by questioning the idea that the intertranslatability between modal

nominalist and platonist theories has ontological consequences. It seems obvious that two

theories could be intertranslatable, and yet quantify over distinct domains. For instance,

one might devise a theory about the depth-charts of two different baseball teams that hap-

pen to have the same number of players assigned to each position. Since both depth-charts

realize the same “structure,” the two theories have the same ontological commitments.

But if the theories have the same ontological commitments, then these two theories are

theories about identical players! This reply ultimately misses the point in more than one

way. What is of interest for Shapiro are theories that are sufficiently formal (in a sense that is

explained in the chapter); when two formal theories are intertranslatable they both realize

the same structure. For Shapiro, the ontology of a mathematical theory is determined by

the structure that it invokes. So two theories have identical ontological commitments only

as far as their shared structure is concerned—no claim follows that their exemplifications

must be identical.

As something of an aside, I consider how the dialectic might proceed if, contrary to

Shapiro’s intentions, the intertranslatability between modal nominalist and platonist theo-

ries unveils modal nominalist theories as no more than mere synonymous paraphrases of

literal mathematical assertions. The thought here is that, if p is a synonymous paraphrase

of q, then p and q both express the same proposition, and thereby have the same commit-

ments. On the other hand, if p and q differ in their commitments, then p is not, in the end,

an acceptable paraphrase of q. In application, the claim is that if modal nominalist theories

are merely re-writes of platonist theories, then they are only acceptable re-writes if they

say the same things as platonist theories. And since platonist theories are committed to

mathematical objects, then so too are modal nominalist theories. My response is relatively

straightforward: Modal nominalist theories of mathematics are not mere paraphrases of

platonist theories.



Returning to the main event, I take a close look at both Michael Resnik’s and Shapiro’s

structuralisms to determine whether there is any reason to grant credence to the alleged

connections between translation and structure and between structure and ontology. What

I discover is a rather surprising pattern: On both Resnik’s and Shapiro’s accounts of struc-

turalism, the fundamental or core claims to be made are the following: It is a (naturalistic)

presupposition of mathematics that various mathematical theories are coherent. Further, that

a mathematical theory is coherent is a sufficient condition for postulating the existence of a

structure of which the theory is a realization—this claim is hypostasized as an axiom of

Shapiro’s structuralism, and is known as the Coherence axiom. For Shapiro, coherence is a

primitive notion, akin to satisfiability, but a notion that is nevertheless not terribly distinct

from the primitive modality Hellman uses. I say these discoveries are surprising for the

following reasons: It is part of Shapiro’s criticism that modal nominalists lack a plausible

means for justifying the modal assertions that figure in their theories. But according to

Shapiro, it is perfectly acceptable to treat the coherence of mathematical theories as a

presupposition of mathematics. Given the indistinctness of coherence and primitive logical

possibility, there is no barrier to a modal nominalist assuming, as a presupposition of

mathematics, that mathematical theories are logically possible. So if Shapiro is right, there

is either no important problem for the modal nominalist, or it is a problem that the modal

nominalist and structuralist face equally.

Moreover, the modal nominalist contends that she can get well enough along with

only the assumption that mathematical theories are logically possible; meanwhile Shapiro

must defend the additional thesis that, in mathematics, coherence suffices for existence—

his Coherence axiom. Shapiro claims that his structuralist position is to be preferred

for holistic reasons—that structuralism provides the best or most plausible account of

the mathematical enterprise. Since the Coherence axiom is a component of Shapiro’s

structuralism, it gets justified along with the holistic justification for his structuralist

view. I contend, however, that this holistic justification uses the Coherence axiom, and so



cannot provide independent support for it. Shapiro never explicitly constructs this holistic

argument, but his extant work suggests that structuralism is thought to be more plausible

than its competitors in large part because the competition—including the modal nominalist

views defended in this dissertation—retain the ontological and epistemological problems

facing platonism (including structuralist platonism). But the idea that modal nominalist

theories are so troubled is the very claim for which justification is sought. Thus there

is no independent support for the Coherence axiom, and subsequently, no independent

support for the claim that modal nominalist theories are committed to the existence of

mathematical structures.

So much for Shapiro’s criticism that modal nominalism fails to effect a genuine reduc-

tion in ontology. What about his accusation that only a platonist position can incorporate

the justificatory resources of the model-theoretic reduction of the logical modalities? Here

Shapiro can be thought of as eliciting a challenge to the modal nominalist: For the modal

nominalist to show that she in fact has nominalistically acceptable, non-platonistic re-

sources through which to justify the modal assertions that figure in her theories. I call this

“Shapiro’s Challenge,” and although I take it as a serious threat to modal nominalism, I

contend that this challenge ultimately need not be met.

In chapter three, which will be described in more detail in a moment, I argue that

seeking a nominalistically acceptable reductive base for the modal notions is a fool’s errand.

The modal nominalist should not be troubled by her inability to find a nominalistically

acceptable reductive base for the modal notions because reductive theories of modality

do not provide effective vehicles with which to justify modal assertions. (Thus not even

Shapiro can meet Shapiro’s Challenge!) The upshot is that all accounts of mathematics that

are in some way mediated by modality face the same kind of justificatory burdens that

modal nominalist theories face.

Even for those who do not take platonism’s intractable epistemology as a core motivat-

ing feature for adopting their modal nominalist views, the use of modality should still raise


Introduction 14 Modality and Reduction

eyebrows. For it is evident that the use of modality raises some difficulties that are not

evidently present in non-modal philosophies of mathematics; the modal nominalist needs

some degree of assurance that the new problems her view raises are less quarrelsome than

the problems, whatever they happen to be, she takes herself to be avoiding in rejecting

platonism. Imagine a philosopher who takes the Quine/Putnam scientific indispensability

arguments to provide the best evidence for the existence of mathematical objects. Never-

theless, she is not convinced that mathematical objects exist, and feels compelled to make

a case that mathematical objects are actually dispensable in scientific theorizing. She wel-

comes the labors of Chihara, Hellman, and Field, who she takes to have given promising

accounts of how it is possible to carry out scientific reasoning without quantifying over

mathematical objects. Certain categorical possibility and necessity statements are listed

among the claims of these philosophers. But if, for example, the epistemology of categori-

cal modal assertions is intractable, or she has no way of justifying the modal assertions

made by Chihara, Hellman, and Field, then she cannot be fully confident that these views

succeed in rebutting the indispensability arguments. She would do well to show, then,

that her less-than-complete confidence in certain modal assertions is not something that

should trouble her.

III.3 Modality and Reduction

The force of Shapiro’s Challenge depends strongly on the assumption that the absence of a

nominalistically acceptable reductive account of modality creates a uniquely challenging

problem for modal nominalists, qua modal primitivists—viz., that modal nominalists face

the burden of justifying the modal assertions that figure in their accounts of mathematics.

Is Shapiro warranted in concluding that this problem places a unique burden on the modal

nominalist? He would only appear warranted in drawing such a conclusion under the

assumption that reducing modality provides or constitutes a means for justifying modal

assertions. The goal of chapter three is to argue that this assumption is unwarranted.

Shapiro advertises the objection that modal nominalists cannot justify the modal as-



sertions that figure in their theories as an epistemological objection to modal nominalism.

But this diagnosis seems somewhat premature—the problem that Shapiro identifies for

modal nominalists is that, because they espouse modal primitivism, they lack the means

through which to describe the content of modal assertions. That is, modal nominalists

are apparently in the dark regarding the truth-conditions of the modal assertions they

make. It must be admitted that this would seem to suggest that modal nominalists cannot

unproblematically claim to know that various modal assertions are true, but the problem is

primarily metaphysical, rather than epistemic: The nominalist seems unable to justify the

modal assertions she makes because she cannot in the first place state the truth-conditions

for these assertions. In contrast, via the model- or set-theoretic reduction of the logical

modalities, Shapiro is capable of stating the truth-conditions for assertions about what is

logically possible.

Does the fact that the set-theoretic reduction can provide truth-conditions for assertions

about what is logically possible show that these assertions are justified for those who

endorse the set-theoretic reduction? For insight on how to approach this question I turn

first to the most well-studied reduction of modality—David Lewis’s modal realism, which

reduces the metaphysical modalities to possible worlds.

Lewis’s modal realism envisions a pluriverse of spatiotemporally disconnected uni-

verses which is so vast that any way that a universe could be is a way that some universe is.

These universes are Lewis’s possible worlds, and Lewis analyses metaphysical necessity

as what is true in every one of these worlds, and metaphysical possibility as what is true

in at least one of these worlds. Lewis’s reduction is of interest because it constitutes a clear

example of the idea that a reductive theory of modality, solely in virtue of its reductive

character, does not provide the means for justifying assertions about what is metaphys-

ically necessary or possible. This raises the prospect of applying similar reasoning in

support of the claim that the set-theoretic reduction of the logical modalities, solely in

virtue of its reductive character, does not provide the means for justifying assertions about



what is logically necessary or possible.

Lewis does give some indication of how he thinks it is possible to justify modal

assertions. He claims that commonsense modal intuitions are perfectly justified as they

come and that these intuitions are consequences of a principle of recombination (the idea

that, for any two Humean distinct existences, there is a possible world containing those two

existences). Commonsense modal intuitions, together with the principle of recombination,

provide a window into Lewis’s pluriverse of possible worlds. But the reductive portion of

Lewis’s overall theory of modality—the analyses of metaphysical necessity and possibility

in terms of spatiotemporally disconnected universes—serves only as a rewrite rule for

translating modal sentences into world sentences, and vice versa. The analyses themselves

confer no justification upon lone modal assertions or upon lone world assertions. On

Lewis’s view, commonsense modal intuitions and other various accoutrements do all of the

work justifying modal assertions (and thus, under translation, justifying world assertions).

Thus, the fact that Lewis’s theory of modality is reductive has nothing at all to do with

whether it is possible to justify modal assertions under Lewis’s overall theory of modality.

What has this to do with the set-theoretic reduction of the logical modalities? The set-

theoretic reduction of the logical modalities views the set-theoretic hierarchy as providing

the resources for grounding or constructing all possible models. Though set theory is only

capable of functioning as a representational device concerning certain logical possibilities

(e.g., one would not identify the logically possibility that Larry is a fishmonger with a pure

set), it is nevertheless possible to view set theory as providing the truth-conditions for

assertions about the logical consistency of mathematical theories and about the possible

existence of mathematical objects.

The main point from my assessment of Lewis’s reduction applies with little modifi-

cation: The set-theoretic analyses of logical necessity and possibility in terms of models

serve only as rewrite rules for translating modal sentences into set existence claims, and

vice versa, and as such confer no justification upon lone modal assertions or upon lone set



existence claims. Where does this place Shapiro’s objection to modal nominalism? Much

seems to turn on exactly what it means to justify a modal assertion.

If the complaint is that modal nominalists, qua modal primitivists, cannot justify the

modal assertions they make, in the sense that they cannot determine what it is that makes

these assertions true, then I am happy to concede the point to Shapiro. Only I would point

out that Shapiro is equally plagued by this problem (to whatever degree it is problematic)—

Shapiro is after all required to justify coherence claims, and his notion of coherence is itself

a modal primitive that is indistinct from a primitive notion of logical possibility.

But on the other hand, perhaps the complaint is that modal nominalists, qua modal

primitivists, cannot justify modal assertions in the sense that they lack compelling reasons

for believing that such assertions are true. But to say this would unveil Shapiro’s ante rem

structuralism as special-pleading. Shapiro claims that it is an uncontroversial presupposition

of mathematics that set theory is coherent. Modal nominalists are certainly not blind to the

presuppositions of mathematics—such presuppositions are open to all if they are open to

anyone. Given the lack of any important distinction between Shapiro’s notion of coherence

and a primitive notion of logical possibility, the alleged mathematical presupposition

that set theory is coherent can be appropriated by the modal nominalist as sufficient

evidence for believing that set theory is logically possible. Thus, if Shapiro is justified, via

mathematics, in believing that set theory is coherent, then modal nominalists are justified,

via mathematics, in believing that set theory is logically consistent. Shapiro’s “epistemic”

criticism therefore ends in a wash. The modal nominalist is in no worse of a position than

Shapiro when it comes to justifying modal assertions.

Let me recap. The modal nominalist eschews commitment to abstract objects by

taking on the burden of an increased ideology of primitive modal notions. While this

obviates the metaphysical and epistemological problems invoked by the postulation of

mathematical objects, it would appear to raise new problems on the ideological front. The

modal nominalist seems obligated to explain why it is, that, for example, she is justified


Introduction 18 Naturalism

in believing that set theory is consistent or possible, and it is by no means clear that this

is an improvement. In the case of Hellman’s view, the problem is to justify assertions

about the primitive logical possibility of the existence of models of mathematical theories.

For Field, the problem is to justify assertions about the primitive logical consistency of

conjunctions of axioms of mathematical theories. And for Chihara, the task is to provide

an account of the metaphysical modalities that justifies the constructibility assertions

that figure in his type-theoretic recovery of mathematical reasoning. Nevertheless these

problems are not, in the end, especially problematic for modal nominalists. The inability

of the set-theoretic reduction to provide a means for justifying assertions about what

is logically possible shows that Shapiro’s reliance on set theory ultimately provides no

justificatory benefits over treating such assertions in a face-value, primitive manner. It

follows that Shapiro cannot claim to reside in a position of superiority when compared

to modal nominalism on point of justifying modal assertions—there is no avoiding the

kind of justificatory questions that get raised under modal primitivism. Thus Shapiro’s

Challenge is not uniquely problematic for modal nominalists.

IV Naturalism

The philosophical orientation of naturalism is advocated by many philosophers, and

indeed, by many philosophers of mathematics. There is some controversy over just what

being a naturalist amounts to; every philosopher that calls herself a ‘naturalist’ seems to

mean something slightly (or not-so-slightly) different by the term. Therefore it would

be pointless to seek out the one “true” naturalism. Nevertheless, the fact remains that

two influential advocates of naturalism have advanced characterizations of scientific and

mathematical method that appear to conflict with modal nominalism. These two influential

naturalists are John Burgess and Penelope Maddy.

It might at first seem ironic that anyone should think that nominalism (including

modal nominalism) is inconsistent with naturalism. Is not nominalism partly motivated


Introduction 19 Naturalism

to account for mathematical reasoning and the scientific applications of such reasoning

in ways that do not utilize the kinds of mysterious epistemic faculties often present in

non-nominalistic accounts of mathematics (e.g., under Godel’s platonism)? And is not

nominalism also partly motivated by the idea, present in scientific methodology, that one

should prefer simpler theories? Are these not in some sense scientific reasons for pursuing

nominalism? In a sense, yes. But in another sense, no. One might worry that these are

only pseudo-scientific platitudes that arise from a dangerously coarse-grained perspective

of science and scientific methodology. Both Burgess and Maddy advance something like

this worry as a naturalistic criticism of nominalism. Each is a naturalist in the Quinean

tradition. That is, they both share Quine’s belief that science is the ultimate arbiter of

questions of existence. However, they disagree quite extensively on the details.

Though not treated in this dissertation, a few remarks on Quine’s naturalism are in

order.4 Quine’s contention is that the evidential standards of science are the best means

available for acquiring knowledge about the world. Thus Quine repudiates the Cartesian

“dream” of finding an indubitable rational basis for all knowledge. On this view science—

i.e., natural science—is only to be criticized through the use of its own methodology. What

are the evidential standards implicit in scientific methodology? They are captured by

the well-worn theoretical virtues (or criteria of theory selection): simplicity, familiarity,

scope, fecundity, and agreement with observation. The mathematical existence debate

is no exception for Quine; nominalism and platonism are both subject to acceptance or

rejection based on whether and to what degree they possess the theoretical virtues.

Quine, initially sympathetic to nominalism, eventually came to reject the view and

to admit the existence of mathematical objects. Two of Quine’s other doctrines play an

ineliminable role in explaining why he came to adopt platonism; his holism and his views

on ontology. Quine’s holism holds that scientific theories are only ever confirmed in their

entirety. He includes applied mathematics in his best overall theory of the world. Indeed,

4Here and in the next paragraph I draw from (Maddy 2005b).


Introduction 20 Reflections on Burgess

he is convinced that mathematics is indispensable to the best theory. Therefore applied

mathematics gets confirmed alongside scientific theories. According to Quine’s criterion

of ontological commitment, a theory is ontologically committed to the ranges of its bound

variables. That is, if a theory quantifies over xs, then that theory is ontologically committed

to xs. Quine believes that one’s ontology is determined by one’s best overall theory of

the world. Since the best overall theory involves mathematics, and since mathematical

assertions quantify over mathematical objects, the naturalist is ontologically committed to

mathematical objects.5

Each aspect of Quine’s position has been criticized extensively. And although I too find

many aspects of it disconcerting, I have only described the view so that it might serve as a

backdrop for the ensuing discussion of the naturalisms of Burgess and Maddy. Both views

are outgrowths of Quine’s naturalism, but each departs from it in important ways. And

each has its own unique reasons for objecting to modal nominalism, and in chapters four

and five I respond to them in turn.

IV.1 Reflections on Burgess

Burgess departs from Quine on the status of unapplied mathematics. For Quine, much of

unapplied mathematics is mere recreation. But for Burgess, mathematics—every last bit

of mathematics—is a part of the best scientific picture of the world. Thus the success of

science points to the truth not just of the mathematics that is applied in physical theories,

but of pure mathematics as well.

Burgess’s (2008b) criticism of nominalism (modal and non-modal alike) maintains

that the view is unscientific, and hence inconsistent with the naturalist orientation in

philosophy. On his view, the nominalist can be understood as attempting one of two

projects. In the first case, the nominalist could be advancing a view about the correct

understanding of mathematical language. Perhaps the surface syntax of mathematical

5This paragraph is an outline of Quine’s Indispensability Argument for the existence of mathematicalobjects.


Introduction 21 Reflections on Burgess

language quantifies over mathematical objects, but this need not imply that, deep down,

mathematical assertions are really about numbers and functions. By providing novel

interpretations of mathematical assertions the nominalist hopes to uncover the true content

of mathematics. Burgess calls this the hermeneutic project. In the second case, the nominalist

could be advancing a view about the methodology of science. In reformulating physical

theories in ways that avoid commitment to mathematical objects she hopes to create

a superior, less ontologically burdensome account of the world. Burgess calls this the

revolutionary project.

Burgess argues that both projects are unscientific. The hermeneutic project is unsci-

entific because there is no good linguistic evidence that the nominalist’s interpretations

capture the “true” content of mathematical assertions. The revolutionary project is unsci-

entific because there is good reason to think that the scientific community would openly

reject the nominalized versions of physical theories. Since these are the only two options

available to the nominalist, her peculiar aversion to mathematical objects must be thrown

out as an unscientific (and hence non-naturalistic) dogma.

The obvious reply to make is that none of the nominalist projects defended in this

dissertation can be accurately described as either hermeneutic or revolutionary. But this

reply will not hinder Burgess. Whatever are the best ways to describe nominalist projects,

including modal nominalist projects, Burgess can always offer the fact that scientists

are not engaging in the project of nominalization as good evidence that the eschewing

commitment to mathematical objects is not a going scientific concern and so is not a

principle of scientific methodology. Nominalism, including modal nominalism, is thereby

guilty of being unscientific.

In response I argue that Burgess’s account of what it is to be “scientific” is too vague

to be of any help to him. Burgess gives very little in the way of a positive characteri-

zation of what his naturalism commits him to. He is clear in holding that one is being

unscientific when one openly adopts a methodological principle that runs contrary to


Introduction 22 Reflections on Maddy

one that is uniformly accepted by the scientific community. But it is not clear that this

implies (as Burgess assumes) that one is thereby also being unscientific when one adopts a

methodological principle that, though it is not explicitly used by the scientific community,

nevertheless does not run contrary to any of those that are uniformly accepted by the

scientific community. Modal nominalist philosophies of mathematics are designed not

to interfere with the day-to-day practice of science and mathematics. That in addition

they seek to eschew commitment to mathematical objects is not evidence that they are

unscientific in any damaging sense of the word.

Another problem for Burgess emerges from this discussion. Burgess is a self-described

moderate platonist. Moderate platonism is the view that when scientists and mathemati-

cians accept, without reservation, assertions that appear to quantify over mathematical

objects, that scientists and mathematicians thereby accept or acquiesce to the existence of

mathematical objects. But if scientifically acceptable evidence is required for eschewing

commitment to mathematical objects, then by parity of reasoning it would appear that as-

senting to commitment to mathematical objects similarly requires scientifically acceptable

evidence. I argue that Burgess’s moderate platonism adopts methodological principles

about language and ontology that are not matters of going scientific concern. It follows

that Burgess’s platonism is self-undermining. One cannot at the same time accuse modal

nominalism of being unscientific and fail to realize that a similar fate befalls platonism.

IV.2 Reflections on Maddy

Maddy’s departure from Quine is more extreme.6 She holds with Burgess that Quine’s

naturalism does not do justice to pure mathematics, but she goes further in rejecting

Quine’s holism. Several factors motivate her to abandon this Quinean doctrine. One

factor arises from a case study of the ontological status of atoms in the period before

Perrin’s experiments were widely known. The postulation of atoms enjoyed all of the

theoretical virtues, and yet many scientists persisted in regarding their existence as a

6Here again I draw from (Maddy 2005b).



fiction. For Maddy this is evidence that the mere enjoyment of the theoretical virtues

does not necessarily track the standards of warrant that scientists actually employ. A

second factor involves the recognition that scientific applications of mathematics always

involve some degree of idealization. For instance, the patently false assumption that fluids

are continuous is often indispensable for making predictions, but this moves no one to

believe that fluids are actually continuous. Maddy concludes that scientists do not treat

all theoretical postulations as on a par. A third factor is Maddy’s observation that science

seems not to be done as though the existence of mathematical objects is at stake. This

evidence does indeed suggest that the thesis of holism is misguided and that, even if it

finds a more limited role elsewhere as a principle of scientific methodology, it should not

be employed to confirm the existence of mathematical objects.

Maddy’s naturalism, also known as Second Philosophy, embraces Maddy’s entreaty

that mathematics should be both understood and evaluated on its own terms. The entreaty

to evaluate mathematics on its own terms comes to the idea that mathematical results and

methods should be criticized and supported only on through the use of actual mathematical

methods. That is, mathematical methods and results are immune from non-mathematical

forms of criticism. The entreaty to understand mathematics on its own terms involves

two somewhat stronger notions. First is the idea that metaphysical theorizing about

mathematics must be consistent with the idea that mathematical forms of justification are

themselves sufficient for justifying mathematical claims. Thus, Maddy’s naturalist will

reject theories of mathematics that introduce a justificatory gap between mathematical

forms of justification and metaphysical analyses of mathematical claims. Second is the idea

that metaphysical theorizing about mathematics must be constrained by the methodology

of mathematics. Maddy’s naturalist will reject metaphysical theories of mathematics that

are not fully endemic to the methodology of mathematics.

Maddy’s own position is that mathematical practice is constrained by the facts of

mathematical depth—a collection of purportedly objective facts that serve to distinguish



the mathematical interesting and productive theories and concepts from their mathe-

matically sterile cousins. These facts are purported to comprise the underlying reality

or subject-matter of mathematics. On this position, mathematical forms of justification

(which, Maddy claims, are responsive to considerations of depth or fruitfulness) suffice for

justifying claims about what concepts and theories are mathematically deep, and so these

forms of justification suffice for justifying mathematical claims. Further, this position is

endemic to mathematics in the sense that, at least according to Maddy’s assessment of the

discipline, the facts of mathematical depth constrain mathematical practice, and also in the

sense that mathematical methods are used to reveal these facts.

It stands to reason that Maddy’s naturalist will object to modal nominalism on the

grounds that it violates her entreaty to understand mathematics on its own terms. This

involves pursuing two related criticisms. First, it is not a built-in feature of modal nom-

inalism that mathematical forms of justification suffice (as Maddy insists) for justifying

claims about what concepts and theories are mathematical deep, and it seems probable

that there is a non-trivial justificatory gap between mathematical methods and the modal

nominalist’s metaphysical interpretation of mathematical claims. And second, it would

seem that modal nominalism, given that it is nominalistic in the secondary sense of §II.2,

requires extramathematical resources and motivation—presumably the desire to eschew

commitment to mathematical objects is not a going item on the agenda of mathematical

practice. Thus modal nominalism is not endemic to mathematics. The primary goal of

chapter five is to assess to what extent these objections are damaging to modal nominalism.

But since Maddy has not directly treated modal nominalism, I am forced to turn to her

objections to the views she takes to be inconsistent with her naturalism, as well as to the

arguments she gives in support of the views she takes to be consistent with her naturalism,

in order to evaluate whether it is a serious objection to modal nominalism that it violates

Maddy’s entreaty to understand mathematics on its own terms.

Against the criticism that modal nominalism does not confirm the idea that math-



ematical forms of justification are sufficient for justifying mathematical claims, I show,

using Maddy’s own examples, that this is only a Second-Philosophically objectionable

diagnosis for views that (a) introduce a non-trivial justificatory gap between mathematical

methods and the proffered metaphysical analyses of mathematical claims, and for views

that preclude the possibility of both (b) acknowledging that considerations of fruitfulness

provide an objective constraint on mathematics, and (c) acknowledging that mathematical

forms of justification suffice for justifying claims about what concepts and theories are

mathematically deep. I argue that modal nominalism is objectionable in none of these

ways. Concerning (b) and (c), nothing precludes amending modal nominalism so as to

embrace the idea that mathematical justification responds to objective considerations of

depth—it is possible to account for these things without making the added claim that the

facts of mathematical depth comprise the underlying reality of mathematics. Concerning

(a), I argue that Maddy’s depth-based account of mathematics and modal nominalism are

all equally wanting a propos of an account of logical possibility. Such an account either will

or will not be endemic to mathematics (an observation that is used in my response to the

second criticism of modal nominalism). Either way, Maddy’s depth-based account and

modal nominalists accounts are helped (or harmed) equally.

Against the criticism that modal nominalism is not endemic to mathematical methodol-

ogy, I argue that it is in general not possible to provide a coherent account of mathematics

that is excised of all non-mathematical resources. As Maddy herself acknowledges, mathe-

matical practice is constrained by the requirement of logical consistency. If mathematical

practice is not in the business of justifying assertions about what is logically consistent or

what is logically possible, then that modal nominalists must go beyond mathematics to

justify assertions about what is logically possible cannot be a decisive reason for rejecting

modal nominalism, because everyone must go beyond mathematics to justify these asser-

tions. (Of course, if it is part of mathematics to justify assertions about what is logically

possible, then it is not clear that modal nominalism fails to be contained or sanctioned by


Introduction 26 Conclusions

the methods of mathematics).

That it appears implausible to provide a coherent account of mathematics that is ex-

cised of all non-mathematical resources suggests that Maddy’s entreaty to understand

mathematics on its own terms unveils Second Philosophy as an implausibly strong form

of naturalism. Indeed, the entreaty itself does not appear to be a product of mathematics

but is instead a philosophical proposal about mathematics and its methodology. It would

be special-pleading to insist that Maddy’s entreaty is the only philosophical claim that

naturalists are permitted to make about mathematics. Thus, although I grant that modal

nominalism is inconsistent with Second Philosophy (because it is not endemic to mathe-

matics), nevertheless I do not think that modal nominalism is thereby objectionable—since

modal nominalism is only incompatible with the strongest and least plausible aspects of

Second Philosophy. I close the chapter (and dissertation) with some brief thoughts on

why modal nominalism is compatible with other forms of naturalism that stop short of

embracing the idea that metaphysical accounts of mathematics must be constructed and

motivated using only internal mathematical resources.

V Conclusions

The use of modality in modal nominalist philosophies of mathematics is thought to produce

intractable difficulties regarding the justification of modal assertions. Nevertheless these

difficulties can be seen to be shared by platonists and indeed by all individuals tasked

with justifying assertions about, e.g., which mathematical theories are coherent or logically

consistent, which mathematical structures possibly exist, etc.

The philosophical orientation of naturalism is thought reveal the modal nominalist’s

ambitions as unscientific and therefore objectionable. Burgess’s naturalism depends on

an understanding of what it is to be “scientific” that at best is too vague to pronounce

against modal nominalism, and at worst is self-undermining. Maddy’s focus on the

methodology of mathematics fails to betray modal nominalism as in any kind of conflict


Introduction 27 Conclusions

with mathematical practice, even if her naturalism is capable of sustaining the judgment

that modal nominalism is not endemic to mathematics. Of course, Maddy encounters

difficulties in explaining why it is objectionable when an account of mathematics fails to

be endemic to the methods of the discipline.

The upshot is the establishment of modal nominalism as a tenable nominalist approach

to the philosophy of mathematics. But can anything be said in the way of arguing for

nominalism in the primary sense of §II.2? Though I make no concerted effort to construct

any such arguments, I can at least relay a prima facie justification for nominalism, which

relies on considerations of overall plausibility. Philosophical theories are selected in part

because of their ability to solve or avoid philosophical problems. If two theories T1, T2

share a common set of unresolved philosophical problems {P1, . . . , Pn}, but where T1 has

additional problems {Q1, . . . , Qn}, while T2 has no additional problems, as a general rule of

thumb T2 is preferable to T1. I would like to suggest here that something like this situation

obtains with platonism as T1 and nominalism as T2. Nominalism and platonism both have

unresolved issues concerning how best to account for the practice of mathematics. Both

have unresolved issues concerning how to account for the applications of mathematics in

physical theories.7 And both views must provide an account of how it is possible to justify

modal assertions. However, platonism is saddled with additional unresolved problems

stemming from the postulation of abstract mathematical objects. Hence modal nominalism

has fewer philosophical problems, and is thereby more likely to be true when compared to

platonism.

7The matter of application is particularly delicate. Nominalists like Field completely reconstruct physicaltheories in qualitative terms. Others suggest that the mathematical apparati of platonistic physical theoriesare fictional, while remaining realists about the “nominalistic content” of such theories. Structuralists main-tain that physical objects exemplify mathematical structures. Traditional platonists believe that mathematicaltruths apply directly to physical phenomena. I am not suggesting that all of these views encounter the sameproblem of application. Intuitively the challenge to the traditional platonist is greater than the challengeto the structuralist. Someone like Field is arguably less burdened than someone who utilizes an under-articulated notion of “nominalistic content.” Broad generalizations aside, whether some nominalistic theoryis superior to some platonistic theory must be decided on a case-by-case basis. Examining the subtleties ofall of the various meters of success of philosophical accounts of mathematics is beyond the scope of thisdissertation.


28

Part 1: Modality

There are logicians, myself among them, to whom the ideas of modal logic (e.g. Lewis’s) are not

intuitively clear until explained in non-modal terms.

—W.V. Quine.

The whole idea of possible worlds (perhaps laid out in space like raisins in a pudding) seems

ludicrous.

—Larry Powers

Chapter 1

Modality in the Philosophy of Mathematics

1.1 Introduction

In this chapter I examine in varying detail the three major modal nominalist projects

in the philosophy of mathematics—Charles Chihara’s Constructibility Theory, Geoffrey

Hellman’s Modal Structuralism, and Hartry Field’s fictionalism, discussed in sections two,

three, and four (respectively). What is of interest is that each of these views explicitly

utilizes various kinds of modal assertions to do critical work in eschewing commitment

to mathematical objects. For instance, Chihara argues that mathematical reasoning can

be carried out without reference to mathematical objects by developing a type-theoretic

framework which replaces Russell’s propositional functions with assertions about the

constructibility of open sentence tokens. Hellman and Field each argue that the content of

mathematical theories consists in modal assertions to the effect that various theories or

structures are consistent or possible.


Modality 29 Introduction

Given that each of these three views is vitally involved with modality, it is natural

to wonder whether such involvement constitutes progress toward producing a coherent

philosophical comprehension of mathematics. I show that there are plausible reasons

for supposing that modal nominalist theories do not represent an increase in epistemic

tractability over traditional platonist accounts of mathematics. A concern of note is that

modal nominalists rely on assertions about what is primitively necessary or possible: They

do not offer any reductive analyses of the modal notions they employ, which obscures

the content of the modal assertions modal nominalists make. This raises a question about

how modal nominalists propose to justify claims about what is primitively necessary or

possible. It is far from clear that the modal nominalists are any more justified in making

modal assertions (e.g., asserting the primitive logically possibility of the conjunction of the

axioms of ZFC) than the platonist is justified in making existence assertions (e.g., asserting

the existence of a model satisfying the axioms of ZFC). And the possibility is also raised

that modal nominalists are actually in a worse justificatory position a propos of modal

assertions when compared to platonists a propos of existence assertions.

Although I maintain that the modal nominalist is not in a worse justificatory position

when compared to the platonist, I will not be in a position to fully justify this claim until

the end of chapter three. So for the purposes of this first chapter I am content to note that

modal nominalism raises questions about justifying modal assertions that are, potentially,

at least as serious as the questions the platonist faces about justifying existence assertions:

How does one come to know that various mathematical theories are possible or consistent?

How can one explain why certain theories are consistent, while others are not, all the while

utilizing a primitive notion of consistency? Questions of this tenor form the basis for the

elaborate objection Stewart Shapiro constructs against modal nominalism, which I discuss

in detail in the second chapter, with my reply occurring over the space of chapters two

and three.


Modality 30 Chihara’s Constructibility Theory

1.2 Chihara’s Constructibility Theory

Chihara (1990) develops a philosophy of mathematics that is designed to avoid what he

calls “the problem of existence in mathematics.” Chihara writes that this is the problem

of, “how, in short, [we are] to understand existence assertions in mathematics” (ibid., 3).

According to platonism, existence assertions in mathematics are understood as ordinary

kinds of existence assertions: When one claims that ‘there exist three prime numbers

greater than 7 but less 19’ one is in no uncertain terms asserting the existence of a certain

set of abstract objects (the numbers 11, 13, and 17). But this gives rise to an epistemological

question: If numbers are abstract objects—that is, objects that exist outside of space and

time—then, Chihara asks, “how can the mathematician know that such things exist?” (ibid.,

5). But why suppose that being unable to know that numbers actually exist is problematic?

Chihara explains,

We seem to be committing ourselves to an impossible situation in which aperson has knowledge of the properties of some objects even though thisperson is completely cut off from any sort of causal interaction with theseobjects. And how does the mathematician discover the various properties andrelationships of these entities that the theorems seem to describe? By whatpowers does the mathematician arrive at mathematical knowledge? In short,how is mathematical knowledge possible? (ibid., emphasis added)

Thus, platonism is in epistemological debt. Chihara aims to avoid this burden by showing

how it is possible to carry out mathematical reasoning in a system that does not quantify

over mathematical objects.

1.2.1 The Constructibility Theory

Chihara’s Constructibility Theory is inspired by simple type theory. The type theoretic

approach to mathematics was first promulgated by Bertrand Russell in the wake of his dis-

covery of the famous paradox that bears his name. In type theory, mathematical existence

assertions are associated with claims involving individuals and propositional functions.

Constructibility Theory is a variant of type theory in which Russell’s propositional func-


Modality 31 The Constructibility Theory

tions are replaced with assertions about the constructibility of various open-sentence

tokens. The upshot is that the existence assertions of ordinary mathematical languages

can be replaced with assertions about the constructibility of open-sentence tokens. It is

important to note, however, that Chihara does not intend his Constructibility Theory to

provide an analysis or account of the ultimate meaning of mathematical assertions. Again,

his interest is in bypassing—not in solving—the various problems that, according to him,

undermine the platonist’s view of mathematics.

According to Chihara, instead of asserting that ‘there exists a number satisfying ϕ,’

one can say instead that, ‘it is possible to construct an open-sentence ψ satisfying φ,’

where the open-sentence property φ is a structural analog of the mathematical property ϕ.

Allegedly, and in contrast with ordinary existence assertions, constructibility assertions

are not ontologically committing. Sentences such as, ‘One of Quine’s anthologies is on

the sofa,’ are true only if there exist things like sofas and anthologies of Quine’s work. But

the truths of sentences such as, ‘It is possible to construct a sofa capable of holding one of

Quine’s anthologies,’ do not require the existence of any sofas or of any anthologies.1

The theoretical framework of Constructibility Theory is essentially that of first-order

modal quantification theory, but designed to include what Chihara calls constructibility

quantifiers in place of the modal operators of necessity and possibility. I will here only

describe the syntactic and semantic features of Constructibility Theory that are unique to it.

Regarding syntax, the constructibility quantifiers C, A, function in much the same way

as do the existential and universal quantifiers ∃, ∀. Let ϕ be the open-sentence ‘x is a Quine

anthology on the sofa.’ Then (Cx)ϕ says that it is possible to construct something x such

that x is a Quine anthology on the sofa. Similarly, (Ax)ϕ says that it is always possible to

construct something x such that x is a Quine anthology on the sofa. In this way there is a

1Perhaps coming to understand what such claims mean requires becoming familiar with particular sofasand Quine anthologies, proceeding to form or apprehend general concepts under which these items fall, andthen learning how to determine, in the abstract, what it would take for an object to fall under these concepts.Perhaps in this procedure abstract objects enter the fray at some point. Perhaps—but perhaps not. This issuefalls beyond the purview this dissertation.



functional correlation between the quantifiers C and ∃, as well as between the quantifiers

A and ∀. The constructibility quantifiers are introduced syntactically as follows: If ϕ is

a formula and α is a variable, then (Cα)ϕ is a formula. And, if ϕ is a formula and α is a

variable, then (Aα)ϕ is a formula. Sentences are restricted to those formulas containing no

free variables.2

In order to provide a semantics for the constructibility quantifiers Chihara develops

a modified version of modal quantification theory. He first identifies a K-interpretation

as an ordered quadruple <W, a, U, I> where (simplifying slightly) W is a non-empty set,

a∈W, and U assigns a non-empty domain to each member of W. I is a function that assigns

to constants members of the union of all U(w), where w∈W. I also assigns extensions to

predicates. Chihara notes that in this system, both constants and predicates are treated as

rigid—that is, held fixed across all members of the domain W (ibid., 28).3

Let both M=<W, a, U, I> and M’=<W’, a’, U’, I’> be K-interpretations with β an

individual constant. If M’ differs from M only in what it assigns to β, call M’ a β-variant

of M. Suppose α is a variable, β an individual symbol, and ψ a formula. Then ψα/β is the

result of replacing with β all free occurrences of α in ψ. The semantics for the quantifiers C

and A can now be given. For ϕ=(Cα)ψ, ϕ is true just in case ψα/β is true under at least

one β-variant of M. For ϕ=(Aα)ψ, ϕ is true just in case ψα/β is true in every β-variant

of M.4 The typical notions of consequence, validity, etc., are defined in relation to the

K-interpretations.

On the intuitive picture, W is a set of possible worlds, a is the actual world, U is a

function that assigns domains of sets of individuals to possible worlds, and I assigns, in

each world, extensions to predicates. Chihara is quick to point out, however, that the

intuitive picture is only a picture:

2For the full treatment, see (ibid., 24-7).3Predicate rigidity involves a predicate picking out the same property in every model (or possible world).

This should not be confused with the idea of a predicate picking out the same extension in every model (orpossible world).

4For the full treatment, consult (ibid., 27-39), with important amplifications on (ibid., 55-8).



. . . the above appeal to possible worlds was made to relate the constructibilityquantifiers to familiar and heavily studied areas of semantical research. I,personally, do not take possible world semantics to be much more than a usefuldevice to facilitate modal reasoning. Still, I hope to convince most philosophersby means of such analyses that the predicate calculus I shall be using is at leastconsistent and that it has a kind of coherence and intelligibility that warrantsthe study of such systems. (ibid., 38)5

Anyone familiar with modal quantification theory will recognize the structural similarities

between the introduction of the modal operators for necessity and possibility and Chihara’s

constructibility quantifiers. However, as Chihara is correct in pointing out, the adoption

of the logical framework commonly used in constructing modal logics does not, in itself,

involve commitment to possible worlds in any metaphysically significant way.

In restricting the arena of discourse to open-sentence tokens, Chihara is able to develop

an omega-sorted type theory in which level-0 entities are ordinary, nominalistically ac-

ceptable objects, and level-(n+ 1) entities are open-sentence tokens satisfiable by level-n

entities. It is the objects of level-(n≥1) over which the constructibility quantifiers range. He

describes how to capture cardinality theory, number theory, and real analysis using his

system.6 Cardinality theory is captured by associating level-2 objects with “cardinality

attributes.” For instance, a level-2 object Z is a zero-attribute just in case it is satisfiable

by a level-1 object that is not satisfiable by any level-0 objects. Higher cardinalities are

attained by defining hereditary relations on cardinality attributes. Number theory is then

captured by proving cardinality attribute-correlates of the Peano Axioms. Real analysis

can be captured just as in ordinary type theory, but again with assertions about the con-

structibility of open-sentence tokens replacing Russell’s propositional functions. Rational

numbers can be defined as higher-level open-sentences satisfiable by equivalence classes

of pairs of cardinality attributes, and real numbers as Dedekind-cuts of rationals, etc.

Chihara’s contention throughout is that, in principle, all existence assertions in math-

5Chihara would later see the need to write an entire book to justify this claim, his (1998). More on thisshortly.

6For the details of cardinality and number theory, see (Chihara 1990, ch. 5). For analysis, see (ibid., ch. 6).


Modality 34 Modality in Constructibility Theory

ematics can be replaced by ontologically neutral constructibility assertions.7 Thus he

eschews commitment to mathematical objects via modality.

1.2.2 A Closer Look at Modality in Constructibility Theory

Having described the basic working parts of Constructibility Theory, it should be helpful

to examine in greater detail how Chihara’s use of modality is alleged to eschew commit-

ment to mathematical objects. What does it mean to say that it is possible to construct

an open-sentence token? What kind of “possibility” must one invoke to account for the

constructibility of open-sentence tokens? Is it appropriate to regard the constructibil-

ity quantifiers as ontologically neutral? And does Constructibility Theory represent a

metaphysical and epistemological improvement over platonism? Let me address these

questions in turn.

What does it mean to say that it is possible to construct an open-sentence token?

According to the “intuitive picture” described above,

. . . to say that it is possible to construct an open-sentence of a certain sort is tosay that in some possible world there is a token of the type of open-sentence inquestion. But what would it be for some token of some open-sentence type toexist in a possible world? Here, we can imagine a possible world in which somepeople, who have an appropriate language, do something that can be describedas the production of the token: they may utter something, write somethingdown, or even make some hand signals (Chihara 1990, 40).

So on the “intuitive picture,” what it means to say that it is possible to construct an open-

sentence token is just that there exists a possible world in which this open-sentence token

has been constructed. Of course, Chihara resists claiming that what transpires within

this “intuitive picture” is really what it means to say that it is possible to construct an

open-sentence token. If, then, one cannot depend on the “intuitive picture” to illuminate

what it means to say that it is possible to construct an open-sentence token, what else is

there to say? One might argue that what underpins the constructibility of open-sentence

tokens is akin to what grounds assertability in intuitionistic settings—that there is available7Cf. (Jacquette 2004).



(or describable) an effective procedure for constructing a particular open-sentence token.

But Chihara regards effective procedures as unduly restrictive (ibid., 45-7).

In fact, Chihara has quite little to say in the way of a positive account of what it means

to say that it is possible to construct an open-sentence token. Most of what he has to say is

negative:

. . . (‘It is possible to construct an open-sentence φ such that φ satisfies ψ’) doesnot mean that one knows how to construct such an open-sentence or that one hasa method for constructing such an open-sentence. Hence, the constructibilityquantifier is not at all like the intuitionist’s existential quantifier. Furthermore,it does not mean that one can always, or even for the most part, determine whatparticular objects would satisfy such an open-sentence. . . (Chihara 2004, 172)

Moreover, he is quite willing to deflect further inquiry into the nature of open-sentences

and satisfiability,

. . . my Constructibility Theory is not designed to give us information about howto tell what is an open-sentence or what things satisfy any given open-sentence.It is just not that sort of theory. It has been designed to provide us with a wayof understanding and analyzing mathematical reasoning, in a way that doesnot presuppose the existence of mathematical entities. . . (ibid.)

Chihara can perhaps be forgiven for avoiding a protracted discussion of the nature of

open-sentences and the nature of satisfiability—these are indeed well-studied concepts

and it seems there is little to be gained by examining them in the present setting. However,

the constructibility quantifiers are the foundation of Chihara’s project, and it is startling, to

say the least, that he seems unwilling to directly and positively characterize what it means

to say that it is possible to construct an open-sentence token. David McCarty presses this

point quite forcefully:

For good or ill, I don’t have much in the way of native ken when it comesto possible constructible sentence tokens. I don’t have at my disposal a richconcept of unactualized sentence forms from which to work. There is no ‘folktheory’ of them that I know of. There is little or nothing of interest I am willingto assert with confidence about possible sentence forms. . . To remind us thatChihara’s constructibles are nonexistent only makes matters worse. . . (1993,260)



To better understand the significance of these remarks, recall that Chihara’s principle

motivation is to avoid “the problem of existence in mathematics,” which is understood

in part as the problem of how to understand existence assertions in mathematics. Chihara

does not pretend to be offering an analysis of what mathematical existence assertions really

mean; instead he thinks he can avoid the problem by exchanging existence assertions for

constructibility assertions. But this would appear to give rise to a new problem—that of

how to understand constructibility assertions. Is this new problem worse than the original?

I shall (eventually) argue that it is not, but for the moment I want to continue in the

exploration of modality in Constructibility Theory.

A few brief remarks about the different “kinds” of possibility are in order, if only to

give context to my analysis of what kind of possibility Constructibility Theory requires.

In the metaphysics of modality, it is common to draw a distinction between a number

of different senses of the term ‘possible.’ Standardly distinguished in such discussions

are logical possibility, metaphysical possibility, and nomological possibility. To say that p is

logically possible is to say that p does not lead to a contradiction in classical logic. To say

that p is metaphysically possible is to say that p is consistent relative to the selection of

metaphysical principles. To say that p is nomologically possible is to say that p is consistent

relative to the selection of natural laws. According to the orthodox view, logical possibility

properly includes metaphysical possibility, and metaphysical possibility in turn properly

includes nomological possibility. Another way of understanding this is to say that logical

possibility is basic and captures the “widest range” of possibilities, while holding that

metaphysical and nomological possibility represent important kinds of restrictions on

logical possibility.8

These distinctions are worth raising for a number of reasons. For my purposes, the

most important reason is that in chapter three I shall be concerned with the question

of whether reducing modality provides a means for justifying modal assertions. To the

8This way of distinguishing the different sorts of possibility is controversial. For a different perspective(in particular, on the issue of the proper placement of nomological possibility), see (Fine 2002).



extent that one is interested in reducing the logical modalities, set theory is thought to

provide a reductive base. If Constructibility Theory ultimately relies on logical possibility,

and logical possibility rests on set theory, then a potential circularity arises in which one

eschews commitment to mathematical ontology via modality, but where modality is only

explicable via mathematical ontology. To the extent that one is interested in reducing

the metaphysical modalities, possible worlds are thought to provide a reductive base. If

Constructibility Theory ultimately relies on metaphysical possibility, and metaphysical

possibility involves commitment to possible worlds, then one faces the worry that in

eschewing commitment to mathematical objects one must accept the existence of possible

worlds.

Chihara provides some illumination into the kind of possibility he envisions while

discussing the possibility of constructing open-sentence tokens:

The possibility talked about in the Constructibility Theory is what is called‘conceptual’ or ‘broadly logical’ possibility—a kind of metaphysical possibility,in so far as it is concerned with how the world could have been. (2004, 171).

Thus Chihara is committed to the idea of it being metaphysically possible to construct the

open-sentence tokens described in (Chihara 1990). He takes metaphysical possibility to be

explicable in terms of representations of ways the world could have been (Chihara 1998, 289).

But what, exactly, is a way the world could have been? And why suppose that such a thing is

importantly distinct from a possible world? I hope to uncover answers to these questions

while considering the issue of whether Chihara’s constructibility quantifiers are indeed

ontologically neutral.

Chihara claims that his constructibility quantifiers are not ontologically committing:

The sentences of my theory do assert the possibility of constructing open-sentences, actual or possible. If I say that it is possible to construct a playgroundnext to my house, I am not referring to, or presupposing the existence of, apossible playground. In this sense, I am not “appealing” to a playground thathas never been constructed. (1990, 78).

And this much seems correct—to claim that it is possible for someone to do something



is not to assert that someone actually does something; nor is it to assert the existence of

whatever might be produced were someone to engage in such an action. But analysts have

taken issue with the fact that the constructibility quantifiers depend on an overtly modal

framework—Chihara’s constructibility quantifiers are defined in what is essentially the

modal quantificational system of S5. S5 systems are usually understood as connecting

to modal concepts by way of objects known as possible worlds (hence the association of

modal semantics with possible worlds semantics). The constructibility quantifiers, defined

as they are in an S5 system, thereby appear to inherit commitment to possible worlds. In a

review of (Chihara 1990), Jan Wolenski makes just this point:

Chihara claims that his system is neither nominalistic nor Platonic. However,this standpoint raises serious doubts. Though Chihara does not need to appealto ‘classical’ Platonic objects, his ontology is committed to possible worlds.Thus, one may restate his question in the following form: whether real humanbeings, living today and with the science we have today, ought to believe in theexistence of possible worlds? Nominalists answer no; but Platonists say that itis much better to believe into [sic] the existence of mathematical objects. Andthe whole problem comes back. (1992, 234)

Similarly, Donald Gillies writes in his review:

. . . Chihara’s reduction, though successful, appears to allow in objects, namely:‘possible worlds’, which are just as problematic as those which he seeks toeliminate. Indeed possible worlds seem to me, if anything, more problematicthan abstract mathematical entities. (1992, 269)

He does indeed talk of possible worlds, but this is only a didactic device used tofacilitate the presentation and clarification of the crucial ideas of the system. . . Ido not regard this reply as altogether satisfactory, for it does seem to methat Chihara to make his system work in various respects (to introduce anon-denumerable set of reals for example) is relying on possible worlds as afoundation, and not merely as a heuristic, communication device. (ibid., 270)

Gillies and Wolenski are each making two related criticisms. The first consists of the

accusation that Constructibility Theory, despite Chihara’s brief remarks to the contrary, is

indeed committed to possible worlds. The second is that Constructibility Theory’s com-

mitment to possible worlds creates new, more troublesome problems in the metaphysics



and epistemology of modality. The first criticism, in itself, is not altogether bothersome—

given that Chihara is interested in eschewing commitment to abstract mathematical objects,

and given that possible worlds and mathematical objects are distinct kinds of entities,

that Chihara is committed to possible worlds does not detract from the fact that he has

eliminated commitment to mathematical objects.9 But Chihara is not only interested in es-

chewing commitment to mathematical objects. Recall again that “the problem of existence

in mathematics” is how one is to understand existence assertions in mathematics, which is

partly unpacked as the problem of explaining how mathematical knowledge is possible.

Chihara seeks to avoid rather than solve this problem, by replacing mathematical existence

assertions with the relevant constructibility assertions. But if constructibility assertions are

ultimately elliptical for claims about possible worlds, then coming to know whether the

relevant constructibility assertions are true would seem to require prior knowledge about

what happens in various possible worlds. And here is where the second criticism exerts

its leverage—whether anyone is justified in believing that possible worlds exist and in

believing claims about what transpires within them are questions that are, at best, at least

as quarrelsome as the questions that get raised when contemplating whether anyone is

justified in believing that mathematical objects exist. In effect, Chihara’s “solution” raises

a new problem—of how to justify constructibility assertions (as elliptical for claims about

possible worlds)—that on the surface may be even more quarrelsome than the one he set

out to avoid.

Chihara blocks the second criticism by contesting the first. If he can show that he is

not, in the end, committed to possible worlds, then it would make little sense to complain

that his system inherits the mysteries that surround them. Why suppose that Chihara is

committed to possible worlds? Because Constructibility Theory is constructed using a

modal logical system. However, he argues that there is no reason to suppose that the use

9Chihara writes that, “the fact that the modal notions may possibly carry hidden commitments to abstractentities should not preclude us from investigating the sort of constructivist theory developed in this work”(1990, 180). So, for better or worse, there is a sense in which he does not find this situation problematic.



of such systems involves commitment to possible worlds.

In (Chihara 1998), he goes to great lengths to show that he is able to use possible

worlds semantics as a merely didactic device. His strategy is to provide a novel, Natural

Language (NL) interpretation of modal semantics. The idea is to understand a statement

p’s possibility as expressing that “the world could have been such that p was true,” and

to understand a statement p’s necessity as expressing that “no matter how the world

could have been, it would have been such that p was true.” What, specifically, does

providing NL interpretations accomplish? For Chihara, formal or pure modal (“possible

world”) semantics is just a piece of uninterpreted mathematical structure (ibid., 260).

Modal systems, by themselves, are unable to generate any substantive modal claims.10 For

instance, a formal modal system is unable to say whether it is indeed possible to construct

an open-sentence satisfied by all and only cats in my house. True, modal systems are

given formal interpretations—one can determine, using modal logic, whether there is

some “world” in which a certain set of arbitrarily selected objects falls under the extension

of an arbitrarily selected predicate. But this far from guarantees that one can, salve veritate,

substitute for those objects the cats in my apartment and a particular open-sentence, and

for that predicate the satisfaction relation. NL interpretations are altogether more powerful

in that they are designed to provide what (Plantinga 1974, 127) describes as an applied

modal semantics:

These “interpretations” do more than mathematical structures do: they notonly assign the relevant sort of sets and objects to the parameters of the logicallanguage in question, they also supply meanings or senses to the parame-ters. . . When a logical language is given an NL interpretation, the sentences ofthe language can be regarded as expressing statements that are true or false.(Chihara 1998, 186-7)

As I remarked earlier, the meaning Chihara intends to provide for the modal operators

10Cf. Alvin Plantinga: “To accept the pure semantics, therefore, is not, as such, to acquiesce in anyphilosophical doctrine at all. The pure semantics commits itself to little more than a fragment of settheory. . . The pure semantics does not give us a meaning for ‘�’, or tell us under what condition a propositionis necessarily true, or what it is for an object to have a property essentially” (Plantinga 1974, 127).



involves statements about ways the world could have been. Nevertheless he recognizes

that reasoning in his proposed system is somewhat laborious, and he goes on to expend

a great deal of energy proving his “fundamental theorem,” aiming to show that one

can use uninterpreted S5 structures to draw inferences about his NL interpretation of

modal semantics. Simplifying to a great extent, the idea is to show that the uninterpreted

sentences of possible worlds semantics can serve as adequate representations of what

is true about how the world could have been. The details of this lengthy proof shall be

passed over here.11 What is important is that in doing so Chihara eschews commitment to

possible worlds. It follows that Constructibility Theory is not burdened by the ineliminable

appeal to the metaphysically and epistemologically suspicious entities known as possible

worlds. There is evidence, then, that Chihara can adopt the results of modal logic without

encountering any new ontological commitments. This observation is relevant to addressing

one of Shapiro’s objections (discussed in the next chapter) that nominalists are not entitled

to use model-theoretic reasoning in the construction and advancement of their views. But it

is far from clear that appealing to NL interpretations of modal systems represents a genuine

improvement with respect to justifying metaphysical possibility claims (e.g., claims about

the constructibility of the open sentence tokens required by Chihara’s reconstruction of

type theory).

It seems to me that Gillies’ and Wolenski’s objections, as well as Chihara’s replies, by

and large miss the point. That is because even if it is the case, as I maintain, that Chihara

avoids commitment to possible worlds, that does not automatically provide Chihara with

an account of how it is possible to justify the metaphysical possibility claims made by his

modal reconstruction of type theory. Chihara needs to explain why he or anyone else is

justified in believing that there are ways the world could have been, such that, if the world

had actually gone these ways, there would have been constructed the very open-sentence

tokens that are asserted to be constructible by Constructibility Theory. The question “how

11For the proof itself, see (ibid., 239-59). For an explication of terminology and conceptual resourcesemployed, see (ibid., 182-239).



can the philosopher know that there are indeed ways the world could have been, such that,

if the world had actually gone these ways, there would have been constructed the very

open-sentence tokens that are asserted to be constructible by Constructibility Theory?” is

suspiciously similar to the question about mathematical objects raised earlier by Chihara—

“how can the mathematician know that such things exist?” (1990, 5). Chihara’s concern in

the latter case is that one is committed, “to an impossible situation in which a person has

knowledge of the properties of some objects even though this person is completely cut

off from any sort of causal interaction with the objects” (ibid.) But why is the situation a

propos of ways the world could have been not just as “impossible”? Ways the world could

have been are, presumably, no less causally isolated from humans. How does the modal

metaphysicist discover the various properties of ways the world could have been that

Constructibility Theory seems to describe? By what power does the modal metaphysicist

arrive at modal knowledge? In short, how is modal knowledge possible?

It is one thing to argue that one can use modal semantics to carry out reasoning in

Constructibility Theory (and to show that no commitment to possible worlds arises from

this use of modal semantics), but that is only half of the battle. To say that Constructibility

Theory is thereby cleared of all obligation to justify modal assertions would be no different

than supposing that the platonist is cleared of all obligation to justify existence assertions

by showing that she is entitled to use classical logic. But by doing this the platonist can only

secure non-trivial knowledge about what conditionally exists—knowledge about what

follows from existence assumptions. She would fail entirely to establish the truth of the

axioms of mathematical theories. Likewise, Chihara does not address questions about the

justification for the constructibility claims that function as the axioms of Constructibility

Theory. But without an account here that does the platonist one better, I cannot make sense

of the claim that Constructibility Theory genuinely avoids “the problem of existence in

mathematics”—because it poses another problem that appears to be just as serious—the

problem of what open-sentence tokens are in fact constructible. These are, to be sure,



different problems.12 But the important issue is that they appear to be similar in the degree

to which they threaten the views under which these problems arise. Constructibility

Theory, in order to provide an account of mathematical reasoning that is more plausible

than the account given by the platonist, therefore faces the burden of justifying assertions

about ways the world could have been.13

In certain respects, reductive theories of modality are thought to provide a means

for justifying modal assertions. If p’s necessity consists in p bearing certain (non-modal)

relations to a reductive base R, then knowledge of p’s necessity consists in knowing certain

(non-modal) facts about R and p. Global skepticism aside, if coming to possess the relevant

bits of knowledge about R and p is unproblematic, it would seem reasonable to suppose

that knowledge of p’s necessity is no great mystery. Unfortunately, Chihara is quite open

about not having offered a reductive account of modality:

. . . I am not a foundationalist, attempting to justify all our modal principlesfrom a certain set of self-evident or a priori truths. Nor am I attempting toprovide a semantical foundation for modal logic from within some austerenon-modal framework, such as set theory. Instead, I make free use of suchmodal notions as is expressed by ‘the world could have been such that’, and Ihave no compunctions about appealing to subjunctive conditionals. Thus, I donot attempt to provide analyses of modal notions in the way David Lewis does.(ibid., 207)

However, just because Chihara is not interested in reducing things like ways the world

could have been and subjunctive conditionals to something more basic does not imply that

it is impossible to provide a non-modal account of such things. Such an account would

provide Chihara a way out of the criticism I am raising. Until he is in possession of such

an account, the charge still stands that it is by no means clear that the justificatory problem

Constructibility Theory faces is less onerous than the justificatory problem platonism faces.

Although Chihara does not offer Constructibility Theory as an account of the true12But cf. Shapiro’s objection in the next chapter.13If, contrary to what most platonists and nominalists in the philosophy of mathematics maintain, there

are no significant metaphysical or epistemological difficulties raised under platonism, then (at least withoutfurther argument) neither should Constructibility Theory be thought to raise any significant metaphysical orepistemological questions.


Modality 44 Hellman’s Modal Structuralism

content of mathematics, nevertheless he does offer it as a means for accessing the truths

of mathematics (whatever these truths ultimately consist in) and for carrying out mathe-

matical reasoning. Modal assertions figure directly in his reconstruction of mathematical

reasoning; modality is the foundation of Constructibility Theory. That he appears to incur

the burden of justifying modal assertions is therefore a significant finding. Does a simi-

lar fortune befall Hellman and Field? I shall examine these views in the following two

sections.

1.3 Hellman’s Modal Structuralism

Structuralism in the philosophy of mathematics is a family of doctrines which holds that,

“mathematical theories typically investigate relations holding among items of structures of

a given type in abstraction from the identity of those individual items” (Hellman 1990, 309).

The structuralist position is often linked to Richard Dedekind, who, in the late 19th century,

showed that any two realizations of the Peano postulates are structurally isomorphic (1901,

31-115). Some have taken this to demonstrate that, in the case of arithmetic and number

theory, the mathematical significance of numbers lies not with any particular objects that

happen to satisfy the Peano postulates, but rather with the relations holding between such

objects.14 Similarly, real analysis can be seen as the study of dense linear orders, where

what in particular is being ordered is unimportant. This analysis is thought to generalize

to most, if not all, of mathematics, hence the slogan: mathematics is the science of structure.

Structuralism can be seen in the modern era as stemming from a foundational worry.

Assume that set theory provides an adequate foundation from which to carry out mathe-

matical investigation. This implies that number theory can be constructed in set theory.

Suppose also that one adopts the strongly reductive position that numbers just are sets.

This, it seems, requires one to identify a given number with a particular set. The problem

is that there are infinitely many ways of doing this, and choosing between these myriad

14This position can be contrasted with the views of someone such as Frege who held that the identity andnature of numbers is revealed by self-evident logical principles.



ways is seemingly a pragmatic matter.15 With what, then, is a number to be identified? The

structuralist’s answer is to say that a number is nothing more than a position in a structure;

that a number, in any mathematically significant sense, is not to be identified with any

object whatsoever!16

As described, whatever the merits of the structuralist position are, compatibility with

nominalism is not listed among them. For what is a natural-number structure, or a real-

number structure, if not another kind of abstract mathematical object? Structuralism can

be understood as ontologically economical when compared with platonism; a countable

infinity of natural numbers are exchanged for the natural number structure, an uncountable

infinity of real numbers are exchanged for the real number structure, etc. But to be in

ontological debt to a lesser degree is still to be in ontological debt. Mutatis mutandis, the

nominalist’s metaphysical and epistemological worries about mathematical objects can

be applied to structures just as easily as they can be applied to numbers. How, then,

does Hellman hope to provide a nominalistically acceptable articulation of mathematical

structuralism? According to Hellman, it is possible to get by solely on the assumption that

the relevant kinds of structures are possible:

The core intuition behind the [modal-structural] interpretation is that a sentenceof the ordinary language of number theory (which we may take to be alreadycodified in a suitable quantificational language) states what would hold in anystructure of the appropriate type that there might be, without commitment thatany such structures actually exist. (Hellman 1990, 314)

How is this to be done?17

15For the classical portrayal of this issue, consult (Benacerraf 1965). For doubts about whether this decisionis merely pragmatic, see (Tappenden 2008b).

16This, at least, is the philosopher’s route to structuralism. A more mathematically-sensitive motivationwould have to stem from observing the extent of the fruitful research deriving from increased focus on, e.g.,properties that are common to disparate mathematical structures, rather than any feelings of anxiety aboutthe so-called “multiple reductions” problem just described. (The multiple reductions problem might bethought by philosophers to be particularly salient as an objection to, e.g., Frege’s account of mathematics. Butmathematicians today are generally not troubled by the foundational issues that occupied Frege and manyother mathematicians up through the early 20th century—issues which, for better or worse, neverthelesspersist as important matters of dispute in philosophical circles.)

17What follows is drawn from (ibid.). For the full treatment see (Hellman 1989).



In the case of number theory, one must first define an ω-sequence, that is, a sequence

that satisfies the Peano postulates.

ω-seq(X, f) ≡df [∧PA2]X(sf ) (1.1)

This states that a unary function f over domain X is an ω-sequence just in case X satisfies

f -correlates of the conjunction of the axioms of second-order PA (henceforth PA2) (i.e., the

behavior of the function f mimics the successor function s). Now let A be a consequence

of PA2. That this is so can be given the modal-structural treatment,

�∀X∀f [∧PA2 ⊃ A]X(sf ), (1.2)

which says that, necessarily, if any pair (X, f) is an ω-sequence, then A holds in (X, f). In

order to avoid the vacuous truth of (1.2), Hellman assumes the following:

♦∃X∃f [∧PA2]X(sf ); (1.3)

it is possible for there to exist an ω-sequence.18 With this assumption in hand one can carry

out arithmetical reasoning without assuming the existence of any abstract mathematical

objects. What of real analysis and set theory? Similar treatments are in order. One begins

with an axiomatic characterization of a certain formal system and then asserts the possible

existence of a model of those axioms.19 Thus,

♦∃X∃f [∧RA2]X(<f ), and (1.4)

♦∃X∃f [∧ZF2]X(εf ), (1.5)

18Since the theory in question is second-order, reference to mathematical individuals is avoided. Hellmanoffers several nominalist strategies for explaining how it is possible that the successor-relation obtains in away that generates an ω-sequence (1989, 47-52).

19Hellman notes that such models should be ‘standard’ as opposed to ‘arbitrary’ (ibid., 19).


Modality 47 Modality in Modal Structuralism

where (1.4) is taken to assert the possible existence of a model of second-order real analysis,

and (1.5) is taken to assert the possible existence of a model of second-order ZF set theory.20

1.3.1 A Closer Look at Modality in Modal Structuralism

Hellman is quite open about his system requiring a primitive notion of logical modality.

His critics are quick to pick up on this as well. Stewart Shapiro writes that Hellman,

“demurs from the standard, model-theoretic accounts of the logical modalities. Instead, he

takes the logical notions as primitive, not to be reduced to set theory” (1997, 89). Meanwhile

Michael Resnik notes that, “to avoid introducing abstract objects into his metatheory, he

eschews the possible world semantics and takes modal operators as primitive” (1997,

68). And they are both quick to levy criticisms against Hellman that are analogous to the

criticisms raised above against Chihara—viz., that it is unclear what possible justification

Hellman could have for making the various modal-existence assertions described above:

Even if we eschew an ontology of possible objects, we surely need an epistemol-ogy of possible objects—just as the traditional realist needs an epistemology ofactual abstract objects. How do we know what is possible? No reason is givento think that the modal route is any more tractable than the realist one—andthere is reason to think it is not more tractable. (Shapiro 1997, 228-9)

Let us now turn to the epistemology of modal mathematics. We might expectthat some advantage is to be gained by replacing the question of how we knowthat entities exhibiting mathematical structures exist by the question of how weknow that such objects are logically possible. However, both Field and Hellmanwould be among the first to admit that they have little idea of how we knowthat objects having the structure of an iterative hierarchy of sets are logicallypossible. But they offer us the hope that the epistemology of logical possibilitywill prove more tractable than the epistemology of mathematics. I do not sharetheir optimism. (Resnik 1997, 77-8)

Perhaps Hellman has succeeded in eschewing commitment to abstract mathematical

objects, but in the process he has created new problems regarding knowledge about the20Those familiar with set theory will recognize that the ZF axioms do not exhaust the possible axiom-

atizations of set theory. Indeed, much interesting and fruitful research requires stronger axiomatizations.That is no problem for Hellman, however, as he can appeal to alternative modal-existence principles. It isbest, then, to think of (1.5) as providing the basis for a schema of set-theoretic modal-existence postulations:♦∃X∃f [∧ZF2 + Y ]X(εf ), where one can substitute in for Y any (consistent!) strengthening of ZF.



possible existence of mathematical structures.

What kind of evidence, if any, can be used to support the various modal-existence

assertions described above? Regarding (1.4), Hellman says that it, “is a strong assumption

not reducible to a claim of formal consistency,” though it, “has its roots in our geometric

experience,” but in the end it, “must be regarded as a working hypothesis of classical

mathematics, not as a self-evident certainty” (1989, 45). And regarding (1.5), Hellman

claims that it, “functions as a working hypothesis of realist set-theoretic practice, just as

the possibility of ω-sequences or of separable ordered continua function, respectively, in

the practice of number theory and real analysis,” but (1.5), “is much less directly tied to

experience than these latter modal-existence assumptions, and is, in this respect, far more

speculative” (ibid., 71-2). Eventually he admits that, “how best to describe and assess our

‘evidence’ for such a hypothesis remains one of the most difficult challenges confronting

mathematical epistemology” (ibid., 72).

These remarks suggest that Hellman takes mathematics to be an important arbiter in

deciding which structures are of mathematical interest.21 But admittedly, his description of

these modal-existence claims as “working hypotheses” of mathematics does not amount

to a substantive account of what justifies these modal-existence claims.22 One might,

of course, argue that the fact that mathematicians are willing to carry out derivations

from a particular set of axioms is evidence for the possible existence of a model of these

axioms. But this would presumably just shift the inquiry over to the question as to what

entitles mathematicians to use these axioms in the first place. Responding here that the

embodied structure is possible, or consistent, would clearly be circular. Another option is

to opt for an inductive justification for the possibility claims—that if, e.g., ZF implied a

contradiction, mathematicians would have uncovered such a fact by now. Some may find

this option unattractive—mathematics is thought by many to be one of the few—perhaps

21Penelope Maddy, in her more realistic moments, proposes that what is of mathematical interest is in facta guide to mathematical existence. See chapter five for elaboration and discussion.

22Thanks to Eric Hiddleston for pointing out several flaws in an earlier formulation of Hellman’s positionon matters.



the only—remaining sources of indubitable knowledge, and for such individuals it would

be unpleasant to discover that mathematical reasoning relies on inductively established

theories. But this might be the unhappy fate that faces mathematical foundations; I will

have somewhat more to say on this issue when discussing Field’s nominalism in the next

section.

A third option—one that Hellman himself endorses—is to argue that these modal-

existence assertions are are indispensable for describing and carrying out mathematical

research (Hellman 1989, 96-7). One might worry that accepting the possible existence of a

mathematical structure because doing so is needed to make sense of mathematical prac-

tice is essentially to postulate possibilities in the same objectionable way that Quine and

Putnam postulated the existence of mathematical objects through their indispensability

arguments. Should this worry prevent Hellman from invoking his modalized version of

the indispensability argument? Hellman’s contention is that the Quine/Putnam indis-

pensability arguments are unsound—not that they are invalid. The purpose of his modal

structuralist theory is to show that mathematical objects are dispensable in mathematics

and science in favor of the weaker premise that modal-existence assertions suffice for

mathematics and science (ibid., 97). Still, to say that possible structures are required for the

truth of scientific and mathematical claims is not to explain in detailed terms how human

beings come to know that it is possible for these structures to exist and to know various

things about them.23

Much as with Chihara, modality plays a foundational role in Hellman’s account of

mathematics. The possible existence of a mathematical structure is predicated on the

primitive logical possibility of the existence of a model exemplifying it, and the success

of mathematical reasoning depends on facts about what necessarily follows from the

axioms that characterize the structure. And just as it is unclear how someone in Chihara’s

23As discussed in the second chapter, Stewart Shapiro claims that it is a presupposition of mathematics that,e.g., set theory is consistent. If philosophers of mathematics are uncontroversially permitted to recognizesuch presuppositions, then Hellman’s justificatory burden does not seem to be terribly great.


Modality 50 Field’s Fictionalism

position is to justify assertions about which open-sentence tokens are indeed constructible,

it is also unclear how someone in Hellman’s position is to justify assertions about which

structures are possible (just as it is unclear how the platonist is to justify assertions about

the truth of mathematical axioms or the existence of mathematical objects). That Hellman’s

modal structuralism appears open to this criticism is therefore a significant finding—for

Hellman, like Chihara, worries that traditional platonist views have difficulty providing,

“a reasonable integration of mathematical knowledge with the rest of human knowledge”

(ibid., 3). It is simply not evident that the difficulties raised by modal structuralism are any

less insuperable than those raised by platonism. Does Field’s fictionalism raise a similar

set of difficulties? I turn now to his his account of mathematics.

1.4 Field’s Fictionalism

Both Chihara and Hellman can be understood as combating the platonist’s supposition

that the soundness of mathematical reasoning and the truth of mathematical theories

require the existence of mathematical objects. Hartry Field addresses a variation on this

theme—to him, it is the prima facie objectivity of mathematics that leads philosophers to

adopt platonism (1998b, 387). Consider, for instance, the undecided status of Goldbach’s

Conjecture (that every even integer greater than two can be expressed as the sum of two

primes). It seems reasonable to suppose that there is a correct (i.e., objective) answer to

the question as to whether Goldbach’s Conjecture is true. Of course this truth has to

consist in something, and if that something is not objective then one risks undermining

the prima facie objectivity of Goldbach’s Conjecture. Abstract mathematical objects are a

perfect fit: they have mind-independent existence and they would appear to settle whether

Goldbach’s Conjecture is true. Thus, the objectivity of mathematics is to be settled by

regarding mathematics as true, and the truth of mathematics is to be explained in part

through reference to abstract mathematical objects.

As motivated, then, platonism can be undermined in two ways. First, one can deny



that the truth of mathematical assertions requires the existence of mathematical objects,

as Chihara and Hellman have done. Alternatively, one can deny that the objectivity

of mathematics is to be settled by regarding mathematical assertions as true. It is this

alternative strategy that Field implements. That is, Field does not regard mathematics as

something that is true—for him, mathematics is just an elaborate fiction.

Field takes as motivation for his program the epistemological problems facing tradi-

tional platonism. He claims that it is not just the causal inertness of mathematical objects

that makes mathematical knowledge so mysterious, but instead the absence of any account

whatsoever for explaining how it is mathematicians come to possess reliable beliefs about

the Third Realm (Field 1989a, 25). He explains:

. . . what causes the really serious epistemological problems is not merely thepostulation of causally inaccessible entities; rather, it is the postulation ofentities that are causally inaccessible and can’t fall within our field of vision anddo not bear any other physical relation to us that could possibly explain howwe can have reliable information about them. (Field, 1989b, 69).

Now it is not clear that in saying this Field has truly extended the epistemological con-

siderations beyond those about the lack of causal connection between human beings and

mathematical objects; surely the complaint that mathematical objects do not bear any

physical relation to humans which could explain one’s reliable beliefs about them is not

appreciably different from complaining that no one has any causal connection to mathe-

matical objects through which to explain the reliability of one’s mathematical beliefs. But I

will not belabor the point—all that is important to note at the moment is that Field develops

his nominalism largely as a measure against incurring the epistemological difficulties that

besiege platonism.24

Field’s projects breaks into two major components. The first component involves the

creation of nominalized versions of physical theories with the aim of demonstrating that it

is possible to carry out scientific reasoning without assuming the existence of mathematical

24For a slightly more mature account of Field’s misgivings about platonist epistemology, see (Field 1989f,230-9).



objects. The second component involves explaining how the use of ordinary mathematical

reasoning is ontologically innocuous, even when applied in science. Modality plays a

critical role only in the second component of Field’s project; for this reason I shall be rather

curt in discussing the first.

Although Field begins work on both components in his well known (1980), the majority

of the book is given over to the development of a nominalistically acceptable version

of physical theory. With the assumption that spacetime is substantival and mirrors the

structure of the four-dimensional real numbers, R4, he has at his disposal a remarkably

powerful basic ontology of spacetime points from which to construct physical correlates

of mathematical objects and concepts. He produces a nominalized version of Newtonian

Gravitational Theory in flat spacetime. The success of this component of Field’s project

has been called into question on three important fronts. First, it is unclear whether Field’s

strategy for nominalization can be extended to cover more mathematically complicated

physical theories, such as theories of quantum mechanics.25 Second, some have questioned

whether helping oneself to an ontology of spacetime points is nominalistically acceptable.26

Finally, some have questioned whether Field’s use of the “full logic of Goodmanium

Sums”—what allegedly amounts to second-order logic—is consistent with nominalism.27

But since I am interested in how the modal notions play a role in modal nominalist

philosophies of mathematics, and these issues do not directly involve a dispute over

modality, I shall not be concerned to discuss possible resolutions to the problems just

raised.

I turn, now, to the second major component of Field’s work—his account of how

mathematics can be useful even though it is not true. Technical complications aside, Field’s

25This complaint is made quite often, but the locus classicus is (Malament 1982). Responses have been madeon Field’s behalf. Mark Balaguer has given some indication about how to nominalize quantum mechanicsin (Balaguer 1998, 113-27). Mary Leng suggests that caution should be exercised in drawing metaphysicalconclusions from quantum mechanics until the scientific community agrees on what parts of the theoryare just mathematical formalism and what parts correspond to physical reality (2010, 57-75). Malament’scriticism is certainly not the knock-down criticism many take it to be.

26See, for instance, (Resnik 1985a). Field’s response can be found in (Field 1989e).27See (Resnik 1985b, § 3).



central thesis is that mathematics is always conservative. Semantically, this can be explained

by saying that,

Let S be a mathematical theory and let N be a nominalistic theory. Then N + S is a

conservative extension of N.

The idea here is that using a mathematical theory does not allow one to say something

about the physical world that one could not already say (perhaps in a long-winded

fashion) in a nominalistic language. The notion of conservativeness can also be explained

deductively,

Let S be a mathematical theory and N be a nominalistic theory, and let N + S ` A.

Then N ` A.

That is, any physical consequence of a nominalistic theory plus mathematics is a con-

sequence of the nominalistic theory alone. Mathematics, while useful for facilitating

inferences, is strictly unnecessary.28 Of course, Field recognizes that it will not do to merely

state that mathematics is conservative over nominalistic physical theory. It is by no means

obvious that mathematics is conservative; a proof of this result is required. Unfortunately

for Field, to prove that a theory N + S is a conservative extension of a theory N requires the

use of metalogic, and, as it is ordinarily understood, metalogic is riddled with mathemati-

cal objects (e.g., models and proofs). Thus to prove the conservativeness of mathematics

requires having knowledge about the relevant pieces of metalogic, which in turn requires

knowledge about some mathematical objects.

Field avoids this result by invoking modality. Instead of showing that a mathematical

theory is consistent, for instance ZF, by showing that it has a model, and then proceeding

to construct the conservativeness result, Field thinks that it is just as well, as far as consis-

tency is concerned, to show that ZF is logically possible. And the logical possibility of the28Shapiro shows that mathematics is actually not conservative in the deductive sense (1983, 521-31). His

strategy is to show that since arithmetic can be constructed in Field’s physical theory the results of Godel’sfirst incompleteness theorem apply; there is a (Godel) sentence of the arithmetical submodel of Field’sphysical theory that cannot be proven in the theory. For Field’s response, see (Field 1989d).


Modality 54 Modality in Field’s Fictionalism

conjunction of the axioms of ZF (♦AXZF) does a fine job in this regard. What this suggests

is that one can understand claims about the semantic consistency of mathematical theories

as modal claims about the logical possibility of the conjunction of their axioms. This frees

him from commitment to mathematical objects in the production of proofs of the semantic

conservativeness of mathematics. Similarly, by invoking certain modal assumptions, Field

hopes to show that he can use proof theory in an ontologically innocuous way when

securing his syntactic conservativeness result.

1.4.1 A Closer Look at Modality in Field’s Fictionalism

It should be clear from the foregoing that the use of modality in Field’s fictionalism

is restricted to metalogic. The issue that must be addressed now is whether his use

of modality presents any clear justificatory hurdles. There are two components to this

discussion. The first involves saying a little more about how Field proposes to “modalize”

the notion of provability and the semantic notions of consistency and implication. The

second involves examining whether it is easier to justify assertions involving these modal

notions than it is to justify ordinary, platonistic mathematical existence assertions.

Before setting out in earnest it is worth noting that Field is, at least early on, somewhat

reluctant in his use of modality, much like Quine was reluctant in his acceptance of the

existence of abstract objects. He writes:

. . . I do not entirely welcome the introduction of the notion of possibility eveninto metalogic; it seems much less bad than either the introduction of thisnotion into physics or the introduction of pieces of platonic protoplasm withno causal connection to us or to anything we observe and existing outside ofspace-time, but still it is something I would prefer to avoid if possible. (Field1989b, 77)

Again the notion arises that the decision between accepting an ideology of modal notions

and accepting an ontology of mathematical objects is something of a tradeoff; obviously

Field is disposed to think of the choice of modality as less onerous. (He remarks on the

same page that invoking possibility is the “best position available.”) Despite this reluctance,



Field thinks that the modalized versions of the metalogical notions have independent

appeal.

To say that ϕ is provable (or deducible) from a set of sentences Γ is ordinarily under-

stood to mean that there exists a proof (or deduction) of ϕ from Γ. But according to Field,

this treatment of provability is artificial, for it is

. . . more natural to characterize the provability of a sentence in terms of thepossible existence of a string of symbols that meets the condition of being aproof of it, than in terms of the actual existence of an abstract sequence ofabstract analogues of those symbols. (ibid., 76)

I am not certain that this is a very persuasive argument for taking provability to be a modal

notion. That it is possible for there to be a string of symbols that meets the condition of

being a proof that ϕ is undoubtedly a sufficient condition for it being provable that ϕ, but

some might say this only calls for an explanation for why such a thing is possible in the

first place. The platonist at least has the beginning of an explanation—she can point to the

existence of a particular “abstract sequence” as a grounding for this possibility. (Of course,

if one asks for an explanation about why that particular “abstract sequence” exists, she

will quickly get into trouble, too.)

Similar remarks apply concerning why the semantic conception of consistency, ordinar-

ily associated with truth in a model, should be taken to be a modal notion:

. . . isn’t the semantic consistency of the theory of discrete linear orderings withno last element more naturally thought of in terms of the possible existence ofentities that are ordered in the way this theory says, rather than in terms ofthe actual existence of an ordered pair whose first member is an infinite setand whose second member is a subset of the Cartesian product of that set withitself? (ibid.)

As before, it is unclear whether someone not already convinced of the wisdom of under-

standing semantic consistency modally would be won over by these comments. Why

suppose that the modal versions of provability and consistency are “more natural” than

their platonistic counterparts?



The semantic notion of implication holds that a set of sentences Γ implies ϕ when ϕ

is true in all models in which every sentence of Γ is true. Against the standard model-

theoretic account, Field advances the following “simple-minded” consideration:

Suppose someone were to assert each of the following:

(a) ‘Snow is white’ does not logically imply ‘Grass is green.’

(b) There are no mathematical entities such as sets.

Such a person would not appear to the untutored mind to be obviously con-tradicting him- or herself. . . but of course (a) and (b) would be in obviouscontradiction if ‘logically imply’ was defined in terms of sets. . . (1989a, 33-4)

Field takes this consideration to show that the model-theoretic account of implication fails

to deliver the meaning of the ordinary notion of logical implication, even if, by accident,

it is extensionally adequate. As before I am not certain what to make of this argument.

A platonist would not be much bothered by the inconsistency of (a) and (b)—she might

even argue that (b) is necessarily false. That (a) and (b) seem not to be contradictory she

can explain as an accident of psychology or of philosophical prejudice.

So much for Field’s reservations about the platonistic metalogical notions. How does he

explain the sense in which implication, consistency, and provability are to be understood

as modal notions? First, the way to learn about the modal operators �, ♦, is by learning

the rules governing their manipulation. Knowledge of ♦, for instance, consists mainly in

knowledge of certain patterns of inference (e.g., A→ ♦A). Moreover, Field claims that the

only thing special about the modal operators is that they have the various rules governing

their use that they in fact have. But if endorsing certain patterns of inference is the only

thing special about the modal operators, then there is a sense in which any other kind of

logical device that licenses the same patterns of inference ought to count as modal, too.

The function of consistency in metalogic is analogous to that of possibility in modal logic,

just as the function of implication in metalogic is analogous to that of necessity in modal

logic (ibid., 34-5). Hence consistency and implication “fit the pattern” for being modal



concepts. Regarding how one comes to possess conceptual knowledge about consistency

and implication, Field writes:

. . . there are “procedural rules” governing the use of these terms, and it is theserules that give the terms the meaning they have, not some alleged definitions,whether in terms of models or of proofs or of substitution instances. (1991, 5)

On the connection between metalogic and modality, he says:

. . . there is no need to explain this operator in other terms, any more thanthere is a need to explain the negation operator or the existential quantifier inother terms; instead we should regard the modal operator as simply a logicalprimitive, one that we come to understand not by defining it but in whateverway we come to understand the other logical primitives. (Field 1989b, 76)

And later:

I am not supposing that we have an independent notion of possibility clearerthan the operator ‘it is consistent that’, and using this allegedly clearer notionto help illuminate consistency. Rather, I am saying that the consistency operatoris clear on its own, and needs no explication. In calling it modal I am simplyobserving that the laws governing it include the familiar modal principles.(Field 1991, 9-10)

Rather than reducing consistency to possibility, Field sees these notions as equivalent. There-

fore, for Field, knowledge of semantic consistency just is knowledge of logical possibility,

and vice versa.

Field’s position on provability is more delicate.29 He does not attempt to produce a

nominalized proof theory, but instead maintains an instrumentalism regarding platonistic

proof theory. According to Field, there are primitive modal facts about what can be proven

in a mathematical theory. These facts point to the necessity of conditionals of the following

form:

AX ⊃ there is a proof of ¬P in F, (1.6)

Where AX is the conjunction of the axioms of a theory and F is a proof theory. Thus the

29In the following I draw from (Field 1989c).


Modality 58 Field and Justifying Modal Assertions

schema,

�(AX ⊃ there is a proof of ¬P in F ). (1.7)

This, then, serves as modal/proof-theoretic evidence for the impossibility of P . For

instance, suppose that there is a proof of ¬S in ZFC. Then,

�(ZFC ⊃ there is a proof of ¬S in F ). (1.8)

Under the assumption that ZFC is consistent, it follows that it is possible for there to be a

proof of ¬S in F , which Field offers as strong evidence that ¬♦S. Thus, platonistic proof

theory can be shown to be useful in acquiring knowledge about possibilities by means of

modal assumptions that are strictly weaker than those required by the platonist (e.g., the

consistency of ZFC versus the truth of ZFC).

With the connections between metalogic and modality finally forged, it is now possible

to address the important justificatory concerns that Field’s account of metalogic raises.

1.4.2 Field and Justifying Modal Assertions

As can be seen, the association of logical knowledge with modal knowledge performs

important work for Field. It is only after metalogic has been purged of any trace of

platonistic ontology that Field can claim to have eliminated the epistemological problems

facing platonism. Nevertheless, his solution involves primitive modal and logical devices.

Field believes that conceptual knowledge about these primitives can be acquired by

learning the “procedural rules” governing their use. Already, however, there is reason to

question the approach of eliminating logical ontology by appealing to primitive logical

concepts. John Burgess complains that, “[n]ominalism threatens to become trivial and

uninteresting if one is allowed to help oneself freely to primitive operators without any

obligation to explain in more familiar terms what they are supposed to mean” (1993, 182).

I suspect Burgess is worried about a potential slippery slope—if modal nominalists can

get away with eschewing commitment to mathematical objects by appealing to primitive



modal concepts, what in principle prevents someone from adopting primitive notions to

do away with other kinds of objects? On this picture, the modal nominalist appears to be

advocating theft over honest toil. In response to this I would first like to stress that Field

in fact agrees with Burgess about the slippery slope; he is in fact reluctant to appeal to

modality. Field’s concern is not so much that to do so would be cheating, but rather that the

decision to “modalize away” a kind of entity is arbitrary—one can modalize away physical

objects just as easily as one can modalize away mathematical objects.30 What entitles

Field to conclude that utilizing modality in metalogic does not set one down the slippery

slope into modal oblivion is his claim that the metalogical concepts are inherently modal,

whereas material objects are not inherently modal. While one could dispute this claim, it

nevertheless suffices as a principled distinction for determining what can and cannot be

modalized away. As a second response to Burgess, I should think that similar remarks

apply to the platonist as well: If it is wrong to invoke primitive notions whenever one

encounters a philosophical problem, why is it not for the same reason wrong to postulate

the existence of some kind of object whenever one encounters a philosophical problem?

Are not both of these strategies equally ad hoc? Why, therefore, is platonism not also “trivial

and uninteresting,” helping itself freely to abstract objects without any explanation as to

how one can have knowledge about them?

I insist that if there is anything wrong with Field’s strategy, it is not because his method

for eliminating mathematical objects in metalogic is an instance of some generally suspect

procedure. Rather, if his view raises any serious justificatory problems, it must have to do

with the particular kinds of primitives he uses. In addition to becoming familiar with the

“procedural rules” for the logical primitives, has Field any account of how it is possible to

justify assertions about what is logically possible or consistent? In particular, does Field

have the resources to justify assertions about the logical consistency of conjunctions of

axioms of important mathematical theories? Unfortunately, Field’s answer is that, “neither

30See (Field 1989f, 252-68).



I nor anyone else that I know of has a great deal to say about the epistemology of modal

claims” (1989d, 140). Again, to counter any reductive means one might have for explaining

the logical primitives, he later offers the reminder that logical possibility, “is not something

to be explained in terms of entities (e.g., models, formal derivations or possible worlds)”

(Field 1989c, 86). What hope is there then for justifying assertions about what is logically

possible? Field sees this as an epistemic issue:

Doubtless the epistemology of assertions of logical impossibility and logicalpossibility needs developing, and there are serious problems to be overcomein doing so; but I think that they are clearly different epistemological problemsthan the epistemological problems that are most characteristic of mathematicsand that motivate anti-realist positions. (Field 1989f, 252)

According to Field, the hopes are dim for an unproblematic epistemology for modality.

However, this should not trouble the modal nominalist too greatly on account of the fact

that the epistemological problems she faces about logic and modality are different from

those facing platonism.31 As I shall explain in chapter three, it is not entirely clear that

this is primarily an epistemological problem. But granting for the moment that it is an

epistemological problem, why should it be supposed that, on account of it being a different

epistemological problem from the epistemological problem facing platonism, that it is

thereby a more tractable problem? Could it not be the case that, all things considered, the

epistemological problems facing logic and modality are at least as troublesome as (or even

worse than) those facing platonism?

According to Resnik, it is unclear that Field has improved the situation:

There are too many unanswered questions and objections on the books con-cerning the scope of logic and the notion of logical possibility for me to believethat anyone can be more justified in appealing to modalities than he would bein appealing directly to abstract entities. (Resnik 1985b, 207)

31Of course, if Field is right about mathematical knowledge being logical knowledge, then it hardlymakes any sense to say that the problem of logical and modal knowledge is different than the problem ofmathematical knowledge. The proper interpretation of Field’s passage is that the epistemological problems oflogic and modality, as the modal nominalist understands logic and modality, are distinct from the epistemologicalproblems facing mathematics, as the platonist understands mathematics.



Resnik, unfortunately, does not make it entirely clear just what questions and objections

make logical modality so dubious. In any event I do not think Field is terribly troubled by

this sort of criticism. I imagine he would understand Resnik to be calling into question

how one acquires conceptual knowledge about the logical modalities, and Field has a reply

to that objection.32 But this kind of conceptual knowledge can only take one so far. For

instance, Field can account for the fact that the consistency of a set of axioms has certain

implications. But he cannot, it seems, account for the fact that a certain set of axioms is

consistent. This prompts Resnik to levy the following criticism:

This leaves us with the question of how nominalists can come to know thepossibilities required for their use of platonist meta-logic and mathematics.Unfortunately, Field has little to say about this. He thinks that there are someobvious principles involving logical form to which the nominalist can appeal.But since their use is clearly limited, he also mentions coming to know possibil-ities inductively (e.g., our experience with ZF might be part of our basis for ourknowledge that it is possible). In the end, however, he admits that neither he‘nor anyone else that I know of has a great deal to say about the epistemologyof modal claims. . . ’ I find this quite ironical in view of Field’s objection tomathematical objects on the grounds that we lack an epistemology for them.(ibid., 204)

It would seem that Field is in a position similar to that of Hellman concerning the justifica-

tion of particular assertions of possibility. In order to clearly demonstrate the justificatory

superiority of fictionalism, he would have to show that it is easier to justify logical possi-

bility assertions than it is to justify mathematical existence assertions. And he certainly

has not shown this. Prima facie, the fictionalist is unable to claim justificatory superiority

over platonism.

In more recent work, Field has attempted to shed more light on the acquisition of

logical knowledge. By construing logical knowledge as a priori knowledge Field thinks

he can explain why it is that logical knowledge does not raise the same epistemological

problems that platonism raises—i.e., that there is no Benacerraf-style epistemological

32His reply, of course, is that one learns about logical modality through the rules governing the use of themodal operators.



problem for logical knowledge.33 However, it is unclear that taking logical knowledge to

be a priori knowledge is at all helpful in explaining how one acquires knowledge about

the consistency of a mathematical theory—e.g., certainly the consistency of ZFC axioms

cannot be justified by a priori insight alone. It is similarly unclear whether Field succeeds

in purging logic of Benacerraf-style epistemological worries.

I qualified the conclusion of the argument at the end of the second-to-last paragraph as

prima facie, and now I would like to explore whether the argument withstands scrutiny.

The conclusion that fictionalism is not superior to platonism concerning basic issues of

justification turns on the assumption that justifying that a mathematical theory is consistent

is no less intractable than justifying that a mathematical theory is true.

Here is a prima facie argument for the conclusion that justifying consistency assertions

(and hence knowledge of consistency), is less suspect than justifying existence assertions

(and hence knowledge of truth). The argument’s main premise is that set theorists have

yet to discover a contradiction in, e.g., ZFC, despite decades of trying.34 This premise is

offered as inductive evidence for the following two claims:

C1. ZFC is true.

C2. ZFC is consistent.

Clearly, (C1) implies (C2), so assertions of consistency cannot be any more suspect than

assertions of truth and existence. The remaining options are that the available evidence

supports (C1) and (C2) equally, or, alternatively, that the evidence favors (C2) over (C1).

Intuitively, when holding p fixed, the claim that p is consistent is weaker than the claim that

p is true. And, holding one’s evidence fixed, inductively arguing for a weaker conclusion

produces a stronger argument, i.e., an argument with a conclusion that is more likely to

33See (Field 1996), (Field 1998a), and (Field 2004).34During a presentation at the 2009 Midwest Philosophy of Mathematics Workshop, the set theorist Hugh

Woodin claimed that no contradictions will be derived from ZFC even after 500 years and that therefore ZFCshould be accepted as true. The following day he increased the time scale of his claim by a factor of ten. Thefinitists in attendance were not pleased by these claims.



be true. (C2) is weaker than (C1), ergo premises one and two inductively support (C2)

more strongly than they support (C1). But then of the two possible conclusions, (C2) is

more likely to be true. This would seem to imply that justifying assertions about what is

consistent is less onerous than justifying assertions about the existence of mathematical

objects and the truth of mathematical theories. End prima facie argument.

Is there anything wrong with this prima facie argument? Certain mathematical struc-

turalists say yes. Indeed, they will call attention to the fact that this argument depends on

the tacit assumption that it is false that (C2) implies (C1). If this assumption is dropped

then of course it no longer follows that there is more reason to believe (C2) given the

available evidence. How can the implication from (C2) to (C1) be justified? It would surely

be a mistake to argue for the general philosophical thesis that possible truth implies actual

truth; that would be to render the distinction between possibility and actuality completely

meaningless. And, in S5, it would lead to the absurd conclusion that everything that is

possibly true is necessarily true. (Of course, one philosopher’s modus tollens is another’s

modus ponens!) But perhaps there is something special about mathematics; perhaps all

one needs to do to prove the existence of a mathematical object is to show that it is possible

for that mathematical object to exist. In the next chapter I will take up in detail several

arguments that suggest that there is indeed something special about mathematical objects

in that their existence can be inferred from their mere possibility.

A different argument a platonist might offer here is that if mathematical concepts were

true of nothing, mathematicians and philosophers would be unable to fasten upon the

refined concepts that they in fact develop (e.g., of function, natural number, real number,

group, etc.). That is, the ease with which mathematicians can conceptualize mathematical

concepts is evidence that they have extensions. I confess to a large degree of skepticism

about this argument—I suppose that there exist no golden mountains in the universe of a

shape and size similar to Mount Everest, but this certainly does not imply that the concept

of such a golden mountain is somehow unconceptualizable. In fact, the concept is very



easy to conceptualize, and moreover, is refined enough to allow someone, in principle,

to determine whether any particular object falls under the concept. If the platonist’s

justification is anything like this, she is obligated to show that mathematical objects are

somehow special in constituting counterexamples to the generally false principle that it is

not possible to conceptualize about concepts that do not have extensions.

One might reply that the example of the golden mountain misses the point, and that

what is really at issue is the ability to form concepts based on well-developed theories.

But that reply will not help the platonist. Consider that the history of science is riddled

with cases of theories that have been rejected on account of the existence of recalcitrant

observations. I presume, for instance, that the plum-pudding model of the atom was

rejected because it failed to account for the results of various experiments and no longer

provided the best explanation for various observed phenomena. But there is little sense

to be had in saying that the model was rejected because there is something inherently

unrationalizable about the concept of a plum-pudding atom. The model fails to cohere with

actual observations, but so does the existence of a golden mountain of a shape and size

similar to Mount Everest. So it is possible to construe a theory as false, even when one has

an “intuitive grasp” of the concepts employed in it; J.J. Thomson surely had a developed

understanding of the hypothesized relationship between the atom and his Christmas day

dessert.

Still, the platonist might object that what is important here is not the development of

theories in general, but that what is at issue is the development of mathematical theories

and mathematical concepts. And she would not be without justification in making this

claim—the development of the calculus, and, later, real analysis, can be understood as

an attempt to come to a greater understanding of concepts like real number, infinite sum,

etc.35 It does not seem unreasonable to explain this period of mathematical history as

an exercise in figuring out what conception of the real numbers is true. If the platonist

35For further discussion on the role of concept formation in mathematics, see (Tappenden 2008a) and(Tappenden 2008b).



is to understand this as the most reasonable conclusion to draw, she must explain why

the drawing of alternative conclusions (e.g., that under certain conceptualizations of

the real numbers it is easier to prove theorems in analysis) is somehow less reasonable.

Nevertheless, even if the platonist is successful in doing this, she will only have succeeded

in showing that sometimes mathematicians reject a mathematical concept because it fails

to be true of anything (of mathematical interest). In all probability, there are infinitely

many (relatively) consistent extensions of ZF, and each such extension would appear to

correspond to a determinate conceptualization of ‘set.’ Many platonists are inclined to

regard these extensions as true of nothing36 and of no mathematical interest. But it hardly

sounds reasonable to say that each such conceptualization is unintelligible. How many of

these conceptualizations must the platonist give up?

Mary Leng agrees with my assessment of this kind of platonist reasoning, writing

that, “our ability to form a concept of objects of a particular kind in no way guarantees the

independent existence of such objects” (2007, 106). However, she thinks the same does not

hold when one is concerned to show that mathematical theories are consistent:

Thus, examination of our concepts of number and of set, and finding ways toenvisage how objects satisfying those concepts could be arranged (e.g. as in theiterative hierarchy) can at least plausibly count as (defeasible) evidence for theconsistency of these notions. At the very least, such considerations suggest thatthe prospects for defending consistency claims on such grounds are greaterthan the prospects of defending claims concerning the face-value truth of thetheories we are considering. (ibid., 107)

And this is indeed a plausible claim. No one, except for those inclined to very strict

conceptions of nomological possibility, would doubt that it is possible for there to exist a

golden mountain of a size and shape similar to Mount Everest (one could even imagine it

constructed on a planet inhabited by some advanced intelligent beings). And if the laws

of physics are really up for grabs, then it is prima facie possible that the building blocks

of molecules more closely resemble Christmas pudding than they do planetary orbits.

36An exception is Balaguer’s “plenitudinous” platonism, examined in detail in his (1998).



Why should the possible existence of the set-theoretic hierarchy be regarded as any more

suspect than the previously entertained possibilities?

I have one reservation about Leng’s reply on behalf of the fictionalist. I am not certain

Leng has shown enough about how one can have knowledge of consistency. By my

lights all she has shown is that there is evidence suggesting that mathematical theories

are epistemically possible. To say that p is epistemically possible for a person S is to say

that p is not (known to be) inconsistent with everything else S knows.37 In general, ‘p

is epistemically possible’ is not equivalent to ‘p is logically possible.’ And what is at

issue is not whether mathematical theories are epistemically possible, but instead whether

mathematical theories are logically possible. Knowledge of the former is not the same as

knowledge of the latter; considering the former as reliable evidence of the latter—even if it

is ultimately defeasible evidence—is to commit oneself to a controversial thesis in modal

epistemology. Controversial as all of this is, Leng’s considerations about mathematical

concepts at least demonstrate a possible way to justify assertions about the consistency of

mathematical theories, and this is welcome news for the fictionalist.

All things considered, it would appear that Field’s account of mathematics raises

unanswered questions regarding the justification of assertions about the consistency of

mathematical theories. But whereas modality plays a foundational role in the work of

Chihara and Hellman, its use is rather limited in Field’s fictionalism. Field only uses

modality in order to establish the conservativeness of mathematics to science and to affirm

the consistency of mathematical theories. His project of producing nominalized versions

of physical theories is left untouched. If this project can be successfully extended to

incorporate contemporary scientific theories, then he can describe physical phenomena

without quantifying over mathematical objects and without invoking primitive modal

37Alternative conceptions of epistemic possibility abound; I would like to say that the details are unim-portant to what I have to say here, except for the fact that some have proposed a link between epistemicpossibility and conceivability, and some have gone on to suggest that, in certain circumstances, conceivabilityis evidence of possibility. If it is acceptable to move from p’s epistemic possibility to p’s conceivability, andthen from p’s conceivability to p’s logical possibility, then what I have to say next might well be false. Formore on the connection between conceivability and possibility, consult (Chalmers 2002).


Modality 67 Recent Developments

concepts. Nevertheless I am concerned in this dissertation to assess the prospects for modal

nominalist accounts of mathematics, even if Field, who takes all mathematical assertions

to state falsehoods, would see this as a fool’s errand. And Field’s account of mathematics

does involve unsubstantiated modal assertions; a significant finding, even if it is not as

injurious to Field as similar results are to Chihara and Hellman.

1.5 Recent Developments

The platonism/nominalism debate has died down in recent years. Although a handful

of novel positions have been presented since the mid 1990’s, including Jody Azzouni’s

notable contributions,38 focus has shifted somewhat more toward paying closer attention

to the practice of mathematics and to the role mathematics plays in scientific explanations.

However, there have been several recent articulations of mathematical fictionalism and I

would like to say something about them.

1.5.1 Balaguer’s Fictionalism

In (Balaguer 1998), Mark Balaguer motivates two philosophies of mathematics. One of

these is a platonist view according to which all possible mathematical objects exist, and it

has come to be known as “plenitudinous” platonism. The second is a fictionalist version of

plenitudinous platonism according to which no mathematical objects exist, and according

to which the acceptability of a mathematical theory can be partially explained by showing

that the theory is consistent. Balaguer argues at length that plenitudinous platonism

overcomes the most decisive objection to traditional platonism, and that his version of

fictionalism overcomes the most decisive objection to traditional fictionalism.39

According to Balaguer, the most important problem facing platonism is the Benacerraf

epistemological argument, “which holds, in a nutshell, that platonism cannot be right

38See, for example, (Azzouni 2004), where he argues that mathematical assertions are not ontologicallycommitting.

39As a general observation, Balaguer argues that the apparent success of both platonism and fictionalismis evidence that there is no fact of the matter about whether mathematical objects exist. My discussion ofBalaguer is cursory and I will not address his general conclusions. Cf. my discussion of a related parityresult, due to Penelope Maddy, in chapter five.


Modality 68 Balaguer’s Fictionalism

because it precludes the possibility of mathematical knowledge” (1998, 21). His under-

standing of this problem comes to the argument that from the assumptions that (1) humans

exist only in spacetime, and that (2) if any abstract mathematical objects existed they would

exist outside of spacetime, if follows that (3) if there exist any abstract mathematical objects,

human beings cannot have knowledge of them. Like Field, Balaguer is not convinced

that (3) only follows from (1) and (2) provided that one also assumes a causal theory of

knowledge. He thinks that as stated, (1) and (2) already provide prima facie evidence for (3)

(ibid., 23). How does plenitudinous platonism avoid this epistemological objection? Well,

if every possible mathematical object exists, then any consistent mathematical theory is

true of some district of the vast platonic realm. Therefore, coming to know that a math-

ematical theory is consistent—an item of logical knowledge, according to Balaguer—is

all the evidence one needs in order to know that certain kinds of mathematical objects

exist. Compare this result with traditional platonism: Traditional platonism assumes that

only certain mathematical objects exist, which means that not all consistent mathematical

theories will have representatives in platonic heaven. Thus one is cut off from a guarantee

that the objects described in a particular mathematical theory actually exist. By moving

from the assumption that some mathematical objects exist to the stronger premise that all

possible mathematical objects exist, the guarantee is recovered. There is still the issue of the

status of Balaguer’s controversial assumption that all possible mathematical objects exist.

He defends this assumption by analogy, arguing that the plenitudinous platonist’s episte-

mological situation with respect to knowing that all possible mathematical objects exist is

just like the external-world theorist’s epistemological situation with respect to knowing

that the physical world exists (ibid., 53-8). I find this analogy highly implausible. But

whether this defense of plenitudinous platonism is successful is ultimately unimportant

for my purposes; I am only trying to give a sketch of Balaguer’s platonism.

Balaguer argues that the most serious problem facing fictionalism (and nominalist

philosophies of mathematics in general) is what he calls the “Fregean Argument” against


Modality 69 Balaguer’s Fictionalism

fictionalism. This is the argument that the best explanation for the applicability of mathe-

matics in science involves regarding mathematics as true, and that the best explanation as

to why mathematics is true is the one given by the platonist (ibid., 95). Balaguer’s response

to this argument largely consists of amplifying and improving the sort of considerations

advanced previously by Field, and I will not spend any time discussing them. And, as

with Field, Balaguer recognizes the important role the notion of consistency plays in the

fictionalist’s account of mathematics; he also understands consistency as a modal primitive.

Against those who claim that primitive modality is not a nominalistically acceptable notion,

he remarks that, “since that notion is a primitive notion, it is entirely obvious that it isn’t

defined in terms of abstract objects, because it doesn’t have any definition at all” (ibid.,

71).40

If Balaguer is right, then modality plays an even more important role in mathematics

than one would have previously thought. If fictionalism is correct, then knowledge about

logical possibility is needed in order to underpin the consistency of mathematical theories.

If plenitudinous platonism is correct, then knowledge about logical possibility is still

required in order to underpin the consistency of mathematical theories, for it is only

through knowing that a mathematical theory is consistent that one can learn that it is true.

This makes deciding between platonism and fictionalism a delicate matter. Nevertheless

the selfsame modal knowledge is the epistemological foundation for both views. What this

implies is that if skepticism about logical possibility is warranted, then the platonist and

the fictionalist are affected equally. There is subsequently no sense at all to be made of the

various reservations voiced earlier in this chapter about how nominalists are particularly

troubled by the justificatory questions raised by their use of modality.41

40Cf. my discussion of nominalism and primitive modality in chapter 3, §6.41This last point is an important theme of the following two chapters.


Modality 70 Leng’s Fictionalism

1.5.2 Leng’s Fictionalism

The most recent and thorough defense of the fictionalist position is found in (Leng 2010).

Under the auspices of naturalistic philosophy she defends two important theses of the

fictionalist program. First, she argues that nothing internal to the practice of mathematics

supports the claim that mathematical theories are true. Second, in response to the indis-

pensability arguments, she argues on naturalistic grounds that confirmational holism and

scientific realism are false. The upshot is that fictionalism is a naturalistically acceptable

position in the philosophies of pure and applied mathematics.

Leng believes that Field’s “mathematical knowledge as logical knowledge” approach

can be extended to the whole of mathematical practice. She uncritically endorses Field’s

arguments for accepting primitive logical notions. However, she expends some effort

in defense of the idea that logical possibility is a primitive notion. She argues that a

set-theoretic reduction of logical possibility is liable to confuse the order of explanation.

The set theoretic reduction of logical modality explains possibilities and necessities as

facts about what sets exist. On this picture, it is impossible for there to exist a set of all

sets because. . . no such set exists! But on the contrary, according to Leng, the fact that it

is impossible for there to exist a set of all sets is something that is explanatorily prior to

the fact that in platonic heaven there is no set of all sets (ibid., 52). Leng concludes that

the set-theoretic reduction of the logical modalities fails as a reduction, and thus requires

a modal primitive. I suspect that this conclusion is unwarranted. But in chapter three

I shall pursue the related claim that the set-theoretic reduction, even if true, does not

provide a means for justifying assertions about what is logically possible, and hence does

not provide an escape route away from the embarrassing kinds of questions that modal

nominalism raises about what justifies modal assertions. But before I do this I want to

come to a greater understanding of exactly why modal nominalists should be troubled

by their use of modality. I take up this issue in the next chapter, where I discuss Stewart

Shapiro’s objections to modal nominalism.


71

Chapter 2

Shapiro’s Challenge to the use of Modality in Nominalist

Theories

I am inclined to think that existence in mathematics is “Consistency + X” but I do not know how

to solve for X .

—Peter Koellner

2.1 Introduction

In this chapter I examine Stewart Shapiro’s criticism of modal nominalist theories of

mathematics. The criticism has three parts, which are described in section two. The first

part involves showing that modal nominalist theories are “definitionally equivalent” to

ordinary platonist mathematical theories in the sense that structure-preserving translations

can be constructed between them. The second part argues that such translations preserve

much more—in particular they preserve both the ontology and the epistemological prob-

lems of platonism. The third part of the criticism holds that platonists have, whereas

modal nominalists lack, the necessary ontological resources (the set-theoretic reduction

of the logical modalities) for justifying and ascribing content to the modal assertions that

figure in modal nominalist accounts of mathematics. Together these three parts purport to

show that (a) modal nominalists face epistemological problems that are just as serious as

those facing platonists, (b) modal nominalists are unsuccessful in their attempts to eschew

commitment to mathematical objects, and (c) concerning modality, that modal nominalists

are particularly burdened by their use of primitive modality.

Section three provides my first response, which challenges Shapiro’s claim that defini-

tionally equivalent theories share an ontology. This discussion serves to expose and clarify

an important element of Shapiro’s criticism—his reliance on the structuralist view that,


Shapiro’s Challenge 72 Introduction

as far as mathematical ontology is concerned, structure is all that matters. In fact what

Shapiro believes is that definitionally equivalent theories have the same structure and are

consequently committed to whatever ontological and epistemological burdens are to be

had in virtue of their structural commitments; later sections are devoted to describing and

criticizing these structuralist conclusions about the ontology of mathematical theories.

Shapiro does not maintain that his formal notion of definitional equivalence preserves

meaning. Nevertheless as an aside I consider what moves are available if one does

assume that the formal relationship which obtains between platonist and modal nominalist

theories preserves meaning. In section four I discuss some arguments to the effect that

modal nominalist theories are mere paraphrases of ordinary mathematical languages and

that paraphrase is incapable of discharging ontological commitments. My response in

section five is to reject this line of reasoning, ultimately because no modal nominalist of

which I am aware is actually attempting to produce a synonymous paraphrase of ordinary

mathematical language.

Returning to the main event, Shapiro’s criticism of modal nominalism is best under-

stood in the context of his overall structuralist position in the philosophy of mathematics.

In section six I begin to explore whether the structuralist position is capable of validating

the connections between structure and ontology that are alleged to raise problems for

modal nominalism. I examine both the views of Michael Resnik and Shapiro. Ultimately

they have very similar things to say about the connection between structure and ontology:

That for a mathematical structure to exist just is for that structure to be possible. Shapiro in

particular speaks of the coherence (a primitive notion akin to second-order satisfiability)

of an implicit definition of a structure (a set of axioms characterizing a mathematical

theory). Shapiro believes that all coherent implicit definitions characterize actually existing

mathematical structures—objects that resemble ante rem universals. This belief is codified

as an axiom of Shapiro’s structuralism: the Coherence axiom.

The success of the first two parts of Shapiro’s criticism turns on whether the Coherence


Shapiro’s Challenge 73 Introduction

axiom can be justified. In section seven I argue that nothing in the practice of mathematics

lends direct support to the Coherence axiom. Though mathematicians may use something

like this axiom to establish the existence (or good standing) of certain kinds of mathematical

entities, this does not lend plausibility to Shapiro’s existential claims about ante rem

structures. He suggests that the Coherence axiom can be justified on account of its inclusion

in a philosophy of mathematics that provides the most plausible explanation of the overall

enterprise of mathematics. However, this plausibility result is available only after the

criticisms of modal nominalism run through, so this holistic justification for the Coherence

axiom is ultimately circular.

Section eight ends the chapter with a discussion of what is to be made of the third

part of the criticism—that modal nominalists are particularly burdened by their use of

primitive modality. This section also lays down a codification of Shapiro’s Challenge,

which comes to the following: For the modal nominalist to explain, on nominalistically

acceptable grounds, why she is justified in asserting the modal claims that figure in her

theories, and further to explain, again on nominalistically acceptable grounds, why she is

entitled to apply the results of modal logic when constructing and applying her theories.

Although a case can be made that modal nominalists have addressed the latter challenge—

e.g, Chihara by way of his natural language interpretation of modal logic (see the previous

chapter for discussion)—nevertheless modal nominalists have seldom bothered to address

the former. Shapiro is correct to claim that modal nominalists lack clear justifications

for the modal assertions that figure in their theories. However, it does not follow that

the modal nominalist thereby resides in a uniquely damaging position. This is because

no one—the platonist included—seems to be any better justified in asserting the relevant

modal claims (including the platonistic counterparts of the nominalist’s modal claims).

I begin to make the case that the set-theoretic reduction of the logical modalities fails to

provide an acceptable means through which to justify claims about, e.g., the consistency of

PA, a case that is completed in the proceeding chapter.


Shapiro’s Challenge 74 Modality and Ontology

2.2 Modality and Ontology

In (Shapiro 1993) and in the seventh chapter of (Shapiro 1997)1 Shapiro examines what

should be made of an apparent tradeoff in the nominalism/platonism debate.2 Platonism

is thought to face intractable epistemological problems stemming from its postulation of

abstract mathematical objects. If mathematics is about a realm of abstract objects, and if

concrete human beings have no direct epistemic access to this realm, then it follows that

human beings do not have knowledge about mathematical objects, and hence do not have

mathematical knowledge. But surely humans do possess a great deal of mathematical

knowledge. Platonists are burdened with the problem of accounting for the apparent relia-

bility on the part of mathematicians to form beliefs about abstract mathematical objects,

without violating any constraints imposed by naturalized epistemology. Thus platonists

may not avail themselves of any purported ability to directly apprehend the mathemat-

ical realm—no use may be made of any mysterious faculties of intuition. Nominalists,

including modal nominalists, say that the benefits (if there be any) of the platonist’s added

ontology come at too high a cost. Nevertheless nominalists, including modal nominalists,

agree that they must say something about mathematics. The particular strategy I defend in

this dissertation is one that requires taking on some measure of primitive modality. Thus,

platonists are burdened by an increased ontology and the epistemological difficulties that

come along with this ontology; meanwhile modal nominalists are burdened by irreducible

modal concepts and apparently unjustified claims about the consistency of mathematical

theories. According to Shapiro, “we are invited to consider a tradeoff between a vast

ontology and an increased ideology” (1993, 456). But, prima facie, the ontological com-

mitments of the platonist and the ideological commitments of the modal nominalist are

incommensurate, and it is subsequently unclear how to assess such a tradeoff. That is, it is

1Essentially a rewrite of (Shapiro 1993); the main arguments remain unchanged, and in many cases theyare repeated verbatim.

2Shapiro prefers the terms ‘anti-realism’ and ‘realism’ in place of ‘nominalism’ and ‘platonism.’ I adoptboth usages in this chapter.


Shapiro’s Challenge 75 The Emperor’s New Epistemology

by no means obvious which burden is greater, or if the burdens are the same, making the

choice between modal nominalism and platonism itself intractably difficult!

Shapiro reaches a series of conclusions that, taken together, imply modal that nominal-

ism is not to be preferred on either ontological or ideological grounds. The first of these

conclusions is that the modal nominalist’s invocation of primitive modality produces novel

epistemological difficulties concerning how humans acquire knowledge about the modal

claims that figure in modal nominalist theories, and further that these novel epistemologi-

cal difficulties are just as troublesome as the epistemological difficulties facing platonism.

At this stage, the ontology/ideology tradeoff is balanced on the epistemological fulcrum,

and if this were the end of the matter, Shapiro would be willing to entertain the idea that

modal nominalism is to be preferred on account of its ontological asceticism. However,

a subsequent conclusion holds that modal nominalists do not succeed in eschewing com-

mitment to mathematical entities. Therefore modal nominalism is not to be preferred on

ontological grounds, leaving the modal nominalist to search after some other medium

in which to limn the superiority of her views. Finally, Shapiro argues that the platonist

resides in a superior situation with respect to modality, for via the set- or model-theoretic

reduction of the logical modalities, she can give content to and provide justifications for the

relevant modal claims, e.g., the claim that PA is consistent, whereas the modal nominalist

appears to incur the consistency of PA as a brute and unanalyzable assertion.

2.2.1 The Emperor’s New Epistemology

A theme from the previous chapter is that it is possible to eschew commitment to math-

ematical objects and thereby escape the epistemological difficulties facing platonism by

appealing to modality in some way. Charles Chihara’s escape route endorses the metaphys-

ical possibility of constructing open-sentence tokens; Geoffrey Hellman’s method appeals

to the primitive logical possibility of the existence of models of mathematical theories; and

Hartry Field’s strategy reads the consistency of a mathematical theory as an assertion of

the primitive logical possibility of the conjunction of its axioms. In the previous chapter I



also suggested that the appeal to primitive modality raises new questions concerning how

humans are supposed to justify and gain knowledge about the various modal assertions

that Chihara, Hellman, and Field make.3 Shapiro recognizes this latter point, noting that

the modal nominalist, “seems to require an epistemology of the actual and possible, and

it not clear that this is a gain” (ibid., 462). Furthermore, modal nominalists, “accept a

primitive notion of possibility, and we are left with very little idea of what this notion

comes to” (ibid., 465; emphasis added). I take these remarks to be consonant with the

results of the previous chapter, and I will not belabor the point. However, Shapiro argues

that a stronger conclusion is warranted—that the modal nominalist faces epistemological

problems that are identical to the epistemological problems facing the platonist.

Shapiro aims to show that the, “epistemological problems facing the anti realist pro-

grammes are just as serious and troublesome as those facing realism. Moreover, the

problems are, in a sense, equivalent to those of realism” (ibid., 456).4 How is this to be done?

I show that there are straightforward, often trivial, translations from the set-theoretic language of the realist to the proposed modal language, and vice-versa.The translations preserve warranted belief, at least, and probably truth . . . Undercertain conditions, the regimented languages are definitionally equivalent, inthe sense that if one translates a sentence φ of one language into the other, andthen translates the result back into the original language, the end result is equiv-alent (in the original language) to φ. The contention is that, because of thesetranslations, neither system can claim a major epistemological advantage overthe other. Any insight the modalist claims for his system can be immediatelyappropriated by the realist, and vice-versa. The problem, however, lies with the“negative” consequences of the translations. The epistemological problems withrealism get “translated” as well. The prima facie intractability of knowledge ofabstract objects indicates an intractability concerning knowledge of the modalnotions, at least as they are developed in the works in question here. (ibid., 457)

Shapiro’s reasoning runs roughly as follows: Begin with an ordinary mathematical as-

sertion p that follows from a certain set of axioms AX . A nominalistically acceptable

3Moreover, these new questions get raised regardless of one’s motivation in reinterpreting mathematics.Modal nominalists who do not view platonism’s defective epistemology as a rallying call are neverthelessrequired to address concerns about the justification of modal assertions.

4Later on the same page he adds that, “[p]erhaps the source of the epistemological difficulties lies in therichness of mathematics itself.”



rendering of the content of p may be obtained by any number of strategies, but here I

restrict my attention just to the major views discussed in the previous chapter—Chihara’s

Constructibility Theory, Hellman’s Modal Structuralism, and Field’s fictionalism. In Field’s

case, p becomes ♦(AX ∧ p), an assertion of the consistency of p together with the axioms

AX (ibid., 462). In Hellman’s case, slightly more is needed. One must state the possible

existence of a model of AX , ♦AX , together with the claim that it is necessary that p fol-

lows from AX , �(AX → p). One is then licensed to assert ♦p (ibid., 466). Still more is

needed in Chihara’s case. One must identify the type-theoretic version of p, say, pt, and

then: exchange all of the quantifiers of pt with constructibility quantifiers, replace any

predication symbols in pt with satisfaction symbols, and replace any mention of types in pt

with open-sentence tokens (ibid., 468). The upshot is a smooth translation scheme from

the language of set theory into the languages of the various modal nominalist theories.

At this point Shapiro suggests that modal nominalism, “faces direct counterparts of

every epistemic problem with realism” (ibid., 462). He explains:

A fundamental problem for the realist is “How do we know φ” or, to bephilosophically explicit, “How do we know that φ holds of the highly abstractontology?” Under the translation, this becomes “How do we know that φ ispossible?” or “How do we know that the conjunction of φ with the axioms ofthe background theory is possible?” (ibid., 463)

This suggestion amounts to the claim, that, for example, any reason one has for doubting

the truth of the axioms of ZF can be recast as a reason for doubting the consistency

of the axioms of ZF. To say otherwise would be tantamount to averring that the major

epistemological problems of mathematics can be solved by inserting modal operators

into mathematical formulae. But Shapiro insists that, “inserting boxes and diamonds into

formulas, or changing the quantifiers, does not, by itself, add epistemic tractability” (ibid.,

474).

The real trouble begins with the recognition that the translations from the platonist’s

language into the modal nominalist’s language can be reversed. In Field’s case, replace



possibility with satisfiability (ibid., 462). In Hellman’s case, replace possibility with satisfi-

ability and necessity with logical truth (ibid., 466). And in Chihara’s case, replace open

sentences with types of the appropriate level, exchange satisfaction for predication, and

change the constructibility quantifiers to existential quantifiers (ibid., 468). The upshot is a

smooth translation scheme from the languages of the various modal nominalist theories

into the language of set theory. Moreover, if a sentence p of set theory is translated into the

sentence p′ of a modal nominalist language, and p′ is then translated into a sentence p′′ of

the original set-theoretic language, p is equivalent in set theory to p′′. According to Shapiro

it then follows that

. . . the fact that there are such smooth and straightforward transformationsbetween the ontologically rich language of the realist and the supposedlyaustere language of the [modal nominalist] indicates that neither of them canclaim a major epistemological advantage over the other. (ibid., 463)

He reasons thusly: The platonism7→modal nominalism translations show that any ma-

jor epistemological problem of platonism is correlated to some epistemological problem

for modal nominalism. The modal nominalism 7→platonism translations show that any

major epistemological problem for modal nominalism is correlated to some epistemo-

logical problem for platonism. Moreover, any important epistemological insights can be

shuffled around as well. Any support the modal nominalist can muster for some par-

ticular modal assertion can be appropriated by the platonist on account of the modal

nominalism7→platonism translations, just as any support the platonist can muster for some

particular set-theoretic claim can be appropriated by the modal nominalist on account of

the platonism7→modal nominalism translations.5

Shapiro observes throughout that modal nominalists must invoke primitive modal

concepts. Platonists, meanwhile, can appeal to the model- or set-theoretic reduction of the

5“. . . unless the realist invokes some sort of (non-natural) direct apprehension of the mathematical realm,any sort of evidence one can cite for believing in a mathematical assertion φ can be invoked by the [modalnominalist] in defence of belief in φ′. . . But given the other translation—[modal nominalism] to realism—doesn’t the reverse apply as well?” (ibid).



modal notions. Necessary truths are those that come out true in every model. Possible

truths are those that come out true in at least one model.6 Shapiro moreover claims that,

. . . once the model-theoretic reduction is in place, the realist has a lot to sayabout logical possibility and logical consequence. It is a gross understatementto point out that mathematical logic has been a productive enterprise. It is notclear that [the modal nominalist] can use the results of model theory, as theybear on his (primitive) modal notion. (ibid., 464)

The upshot, Shapiro alleges, is that modal nominalists are particularly burdened by their

use of modal concepts. Modal nominalists have not provided any rigorous analyses of

the various modal notions employed in their theories. However, modal locutions are

well-entrenched in natural languages. Competent speakers routinely use, believe, and act

upon modal propositions. Nevertheless these, “everyday modal. . . notions, by themselves,

are too vague to support the detailed applications to surrogates of set theory or type theory

as envisioned by [modal nominalists]” (ibid., 475).7 Commonsense beliefs about modality

may be well-enough equipped to speak about mundane modal facts, such as whether

it is possible for Ohio State to win at Notre Dame, but that is a far and distant cry from

asserting the possibility of the conjunction of the axioms of ZFC. What is more,

. . . in practice, our grasp of modal. . . terminology, as applied to mathematicsat least, is very much mediated by mathematics, set theory in particular. Weinherit the language/framework with the connections to set theory alreadyforged—and to use a worn metaphor, we can’t get off the ship of Neurath.Surely, our [modal nominalists] don’t claim that we still have some sort ofpre-theoretic intuitions of these notions, intuitions that remain uncorrupted, orat least unmodified, by set theory. (Shapiro 1993, 475)8

Consider Paul Cohen’s proof that the Continuum Hypothesis (CH) is independent of

the axioms of ZFC. He did this by constructing models. That ZFC±CH is consistent is

thereby knowledge about derivations of set theory—what entitles the modal nominalist to

proclaim this item as knowledge about some primitive possibility claim? And how could

6For a more detailed account of the model-theoretic reduction, see chapter three, §2 and §5.7See also (Hale 1996).8Similar remarks are expressed in (McKeon 2004, 421).



a modal nominalist ever arrive at Cohen’s result without using model theory? Against

Chihara, Shapiro responds that,

. . . we only understand how the constructibility locutions work in Chihara’sapplication to mathematics because we have a well-developed theory of logicalpossibility, satisfiability, etc. And, once again, this theory is rooted in set theory,via model theory. That is the source of the precision. (Shapiro 1993, 470;Shapiro’s emphasis)

Similar remarks apply to Hellman and Field; the manipulation of the modal apparatus

of either view takes advantage of modal-logical inferences best understood in reference

to model theory. One might adopt a fictionalism with respect to model theory, but this

would presumably invoke higher-order inferences, forcing a modal nominalism-platonism

standoff ascending through “a hierarchy of metalanguages” (ibid., 465). Shapiro’s final

judgment on the modal nominalist’s use of modality is that she,

. . . owes us some account of how we plausibly could come to understandthe notions in question (as applied here) as we in fact do, independent of ourmathematics. Without this, it is empty to use a word like “primitive” and,without this, we can’t give a positive assessment of the progress of anti-realistprograms, or even a judgement that they have achieved a balanced tradeoff.(ibid., 475)

Notice that Shapiro speaks of the modal nominalist encountering difficulty in un-

derstanding the modal notions. But it must be recognized that the resources by which

someone comes to understand a proposition p need not be the very same resources that are

responsible for grounding or making it the case that p.9 Shapiro takes it as a given that if the

modal nominalist were right about modality, then modal assertions would have primitive

grounds. For instance, if the modal nominalist is right, then the consistency of PA is a

brute and unanalyzable fact. Nevertheless it appears to be a (relatively) uncontroversial

datum that most mathematicians know what it means to assert the consistency of PA, and

this is a datum for which the modal nominalist can provide no explanation. She might

hope to salvage an explanation via model theory, but if she rejects the purported ontology9Thanks to Susan Vineberg for pressing this distinction.



of model theory she is left with an unexplained correlation between the primitive modal

facts and the facts of model theory (e.g., she seems to have no explanation for why the

model theory reliably tracks the facts about primitive modality). I think it is this kind of

unexplained correlation that drives Shapiro to speak against the plausibility of the modal

nominalist’s reliance on primitive modality. Since the nominalist can give no explanation

of the uncontroversial datum, her view ought to be rejected in favor of a view that can

explain the datum. The platonist offers such a view in identifying assertions of consistency

with certain model- or set-theoretic constructions. So the complaint is, in essence, that

the platonist has explanatory resources that the nominalist lacks, and moreover that any

plausibility the modal nominalist’s explanations might have is (perhaps covertly) owed to

model theory (of a platonist bent).

Shapiro, then, lodges two kinds of complaints against the modal nominalist a propos of

modality. One complaint is that it is something of a mystery as to how modal nominalists

propose to justify the the modal assertions that figure in their theories, perhaps as a

consequence of the fact that modal nominalists have rather little to say about the content

of these claims. Meanwhile, if Shapiro is right, the platonist can appeal to her prior

beliefs about models or sets in justifying modal assertions. For example, the claim that

PA is consistent can be identified with the claim that PA has a model; the existence

of a model satisfying PA then serves as evidence for the consistency of PA. A second

complaint is that modal nominalists have not earned the right to employ the results of

model theory and modal logic to make derivations from their primitive modal premises.

In both cases the complaint is that modal nominalism is importantly impoverished when

compared to platonism. Modal nominalists have only made serious attempts to combat

the second criticism—e.g., (Chihara 1998)’s NL interpretation of modal semantics and

Field’s insistence that the metalogical notions are themselves primitively modal (see

the previous chapter for discussion). But that is only to go halfway toward responding

to Shapiro’s criticism. Modal nominalists still face core questions that stem not from



the simple fact of their use of modal reasoning but instead from the categorical modal

assertions they make—e.g., in Hellman’s case, the claim that it is possible for there to exist

a model of the axioms of second-order PA. What content can a modal nominalist ascribe

to such an assertion? And how could a nominalist justify such an assertion? Shapiro’s

allegation is that the metaphysical resources of platonist model theory provide the most

(and perhaps the only) insightful answers to these questions—something that I contest in

this dissertation. (To anticipate developments from chapter three, I should add that I also

contest the idea that the problem Shapiro identifies for the modal nominalist is, principally,

an epistemological problem. That modal nominalists appear unable to justify assertions

about what is possible would seem to be a symptom of a deeper, metaphysical problem,

viz., that modal nominalists, in espousing modal primitivism, fail to describe the content of

modal assertions. I will overlook this detail for the remainder of the chapter, proceeding

with Shapiro in describing the modal nominalist’s justificatory problems as epistemological,

rather than metaphysical in nature.)

If Shapiro is right, then the modal nominalist’s ploy in no way helps her view gain

tractability over platonism. At best, she faces epistemological problems that are just as

burdensome as those facing the platonist. At worst, she has no story whatsoever to tell

about how human beings can explain and acquire knowledge about the consistency of

mathematical theories, whereas the platonist can present her model-theoretic construc-

tions.10 The epistemological fulcrum is therefore an equivocal measure of the merits of

modal nominalism and platonism. Shapiro believes that the natural next step for the modal

nominalist would be to argue for modal nominalism on account of its more economical on-

tology. Now, as I have indicated previously, I do not myself take this to be the only or most

effective way of arguing for the modal nominalist theories defended in this dissertation.

Rather, my sympathy for the modal nominalist approach comes to the idea that one should

not believe any claim without the right kind of evidence, and there are good reasons

10Of course, wherefore art the platonist’s warrant in the underlying set theory needed to produce themodel-theoretic constructions? Of this, more later.


Shapiro’s Challenge 83 The Emperor’s New Ontology

for thinking that platonists have not provided compelling evidence for the existence of

mathematical objects. Shapiro, as discussed below, views mathematical structures as free-

standing abstract objects akin to ante rem universals; and Burgess, as I argue in the fourth

chapter, approaches mathematical languages with literalist-platonist presuppositions. The

problem is that these resources—Shapiro’s conception of abstract structure, Burgess’s

literalist presuppositions—do not have a plausible source in the practice of mathematics

itself, and consequently, without cogent supplementary reasoning, they ought to have no

purchase on the naturalist’s convictions. Given the naturalistic subtext of this dissertation,

these are important results. The motivation for modal nominalist theorizing, then, arises

from recognizing that (a) mathematics and science internally provide no good reasons

for supposing that mathematical objects exist, and that (b) there is nevertheless interest

in accounting for mathematical reasoning and the content of mathematical theories in a

fashion that coheres with the absence, internal to mathematics and science, of reasons

for supposing that mathematical objects exist. I will have occasion to expand on these

remarks later in this chapter and in the naturalism section of the dissertation. However,

at the moment I am not trying to argue for modal nominalism but instead to deflect ob-

jections to certain modal nominalist views. Modal nominalists do maintain that they are

not ontologically committed to mathematical objects, but appearances can be deceiving;

perhaps they are not even successful in eschewing commitment to mathematical objects in

the first place—a conclusion that Shapiro goes to great lengths to establish.

2.2.2 The Emperor’s New Ontology

According to Quine’s criterion of ontological commitment, a theory is ontologically com-

mitted to the objects over which it quantifies. When the logical apparatus of a theory

is restricted to non-extended first-order logics, one can read the existential quantifier as

apportioning such commitments. The ontological commitments of the traditional platonist

position are clear in this respect; mathematical existence assertions are expressible in

first-order languages. Meanwhile, Shapiro champions a structuralist position according to



which the “logic of mathematics” is second-order logic, which muddies the ontological

waters as far as Quine is concerned.11 Similarly, modal nominalists envision an expansion

of the logical apparatus to include modality. (Hellman’s account of mathematics uses

second-order modal logic—a departure from Quine in two respects!) It is subsequently

unclear, via Quine, to what the various modal nominalist accounts of mathematics are

ontologically committed. This prompts Shapiro to make the following assessment:

. . . we need a new tool to assess the ontology/ideology of a philosophical/scie-ntific/mathematical theory. Quite simply, the Quinean ontology-through-bound-variables thesis fails if ideology is not held fixed, if not [held] minimal.Indeed, the criterion is outright misleading. (Shapiro 1993, 476)

If Quine’s criterion is incapable of delivering a decision as to which position has the most

economical ontology, what else is there to be said on matters? Should one simply turn

one’s back to the epistemological debate and give up on ontology? But, at least as far as

platonism is concerned, the epistemological questions are driven by the view’s abstract

ontology. So the ontological question is propaedeutical.

Shapiro hopes to remedy this situation by concocting a novel measure of the ontological

and ideological commitments of a theory. This remedy is unapologetically inspired by his

structuralist views:

As a first approximation, the proposed criterion of ontology/ideology is this:A theory is committed to at least the structure or structures that it invokes anduses. If two theories involve the same structures or if the systems described bythem exemplify the same structures, then, at least as far as mathematics goes,their ontologies/ideologies are identical. (Shapiro 1997, 238)

So if two theories invoke or exemplify the same structure, then they have identical onto-

logical and ideological commitments. Of course, what is missing from this first attempt are

criteria of structure-identity and structure-individuation; without such criteria it would be

11(Quine 1986, 67) warns against treating quantification over predicates as carrying ontological commit-ment to predicates, propositions, or attributes, instead preferring to treat second-order assertions as ellipticalfor claims about sets of individuals; hence the slogan, “second order logic is set theory in sheep’s clothing.”Thanks to Michael McKinsey for bringing this issue to my attention.



impossible to determine whether any two theories invoke the same or different structures.

Recall also that epistemological problems are alleged not to be lost in the translations

between nominalist and platonist languages. The technical maneuver Shapiro utilizes in

support of this claim is showing that the theories are definitionally equivalent. In precise

terms,

Two theories T , T ′ are said to be definitionally equivalent if there is a function f1from the class of sentences of T into the class of sentences of T ′ and a functionf2 from the class of sentences of T ′ into the class of sentences of T , such that(1) f1 and f2 both preserve truth (or theoremhood if the theories do not haveintended interpretations) and (2) for any sentence φ of T , f2f1(φ) is equivalentin T to φ, and for any sentence ψ of T ′, f1f2(ψ) is equivalent in T ′ to ψ. (Shapiro1993, 479)

What is the significance of learning that two theories are definitionally equivalent?

I propose that definitional equivalence serve as a criterion of the formal strengthof modal and nonmodal theories and. . . that this notion be used as an indica-tion that the intended structures, and thus the ontology/ideology of differenttheories, are the same. If T is definitionally equivalent to T ′, then neither is tobe preferred to the other on ontological/ideological grounds. (Shapiro 1997,242)

It follows that if a modal nominalist theory is definitionally equivalent to set theory (or type

theory), then these two theories have identical ontological commitments; both theories

are committed to the same structure. Thus modal nominalists like Chihara, Field, and

Hellman do not succeed in eschewing commitment to mathematical objects—each must

countenance mathematical structures.

Shapiro has constructed a broad-ranging criticism of modal nominalist theories of

mathematics. That modal nominalist theories are definitionally equivalent to set theory

allegedly shows that (a) modal nominalists face correlates of the epistemological difficulties

facing platonists, and (b) modal nominalists do not succeed in eschewing commitment

to mathematical objects. Moreover, the absence of a nominalistically kosher reductive

account of logical possibility means that (c) platonism is superior to modal nominalism


Shapiro’s Challenge 86 First Reply

with respect to the justification of modal assertions. If his arguments are sound, then he has

seriously undermined the modal nominalist approach in philosophy of mathematics. In

the remaining parts of this chapter I respond to all three developments. After considering

various unsuccessful ways of amplifying Shapiro’s arguments, I conclude that he fails to

show that modal nominalists are unsuccessful in eschewing commitment to mathematical

objects, vitiating claim (b). In particular, I argue that Shapiro’s ontological parity result

is only available once his Coherence axiom is up and running—an axiom that owes its

plausibility in large part to the criticisms described above. And, although I concede

that modal nominalists face justificatory difficulties stemming from their invocation of

primitive modality, I make a preliminary case—to be filled out more fully in chapter

three—for thinking that the modal nominalist’s burdens are no more taxing than those

facing the platonist, countering (c) and undermining the significance of result (a).

2.3 First Reply

My first reply concerns the purported implications of definitional equivalence. Shapiro

offers definitional equivalence as a criterion of structure-identity. Any two theories that

are definitionally equivalent invoke the same structure, and therefore have identical

ontological commitments.

According to John Corcoran, definitional equivalence is characterized in the following

way:

Let DST be a set of (nominal) definitions of the (uninterpreted) primitives ofS in terms of those of T. DTS is likewise defined. Suppose that when DST isadjoined to T, the axioms of S and the definitions of DTS follow. Suppose alsothat when DTS is adjoined to S, the axioms of T and the definitions DST follow.When T and S are so related, I call them definitionally equivalent. (1980, 231)

Do Shapiro’s translations show that set theory is definitionally equivalent to modal nomi-

nalist theories in this sense? One would have to prove as theorems of the modal nominalist

theories (extended to include nominalized set-theoretic primitives), (a) the axioms of set

theory, and (b) the set-theoretic definitions of the nominalized primitives. As well, one



would have to prove as theorems of an extended set theory (a’) the axioms of the modal

nominalist theories, and (b’) the nominalized definitions of the set-theoretic primitives.

What ‘proof’ means here, as far as Shapiro is concerned, is somewhat unclear. Are such

proofs to be constructed in first-order or second-order logic? And what other logical

resources must be made available? Corcoran says that his notion is neutral between the

underlying logic of the theories in question (ibid.). I am not confident whether he means

by this that (i) it does not matter if either theory uses the same or different underlying

logics, or (ii) it does not matter what the underlying logic of both theories are, provided

that both use the same underlying logic.

In any case Shapiro does not endeavor to explain how one is supposed to embed

his translations in the actual theoretical apparatus of either theory. All that Shapiro

requires is that the equivalences (between φ and f2f1(φ) in T , and between ψ and f1f2(ψ)

in T ′) can be appreciated by neutral observers (Shapiro 1997, 225). In order to actually

construct the requisite proofs (on Corcoran’s picture), Shapiro’s translations must be

viewed as interpreting definitions. In the case of a modal nominalist theory, the modal

nominalism7→platonism translations constitute the definitions of set-theoretic primitives

in terms of the primitives of the modal nominalist theory. The result needed is that when

the modal nominalist theory is adjoined with this translation scheme, the axioms of set

theory and the definitions of the primitives of the modal nominalist theory in terms of the

primitives of set theory follow as theorems (and vice-versa for the other direction). That the

set-theoretic axioms follow can be shown simply by translating the appropriate sentences.

This is most easy to see in Hellman’s case. Take the statement of the possible existence

of a model of the axioms of mathematical theory and “translate” it into an assertion of

the existence of a model of the axioms a mathematical theory. Reversing the translations

provides the means for deducing the definitions of the primitives of the modal nominalist

theory in terms of the primitives of set theory. A similar story can be told by starting

with set theory and adjoining to it the set-theoretic definitions of the primitives of the



modal nominalist theory. But if the underlying logics of the theories are different, then

it is prima facie implausible that these translations alone preserve theoremhood (in the

sense that anything that can be proven in one theory can also be proven in the other).

Thus the preservation of theoremhood must be viewed as an additional criterion that such

translations must meet in order to count as definitionally equivalent in Shapiro’s sense.

I am concerned about the degree to which it is coherent to think of Shapiro’s translations

as mere definitional extensions. Are these translations to be thought of as conservative

extensions? That surely cannot be the case, because the variables of the platonist’s theory

(set theory) and the variables of the modal nominalist’s theory range over disjoint sets of

objects. But then in what sense is the reader to believe that Shapiro’s translations capture

the set-theoretic primitives in terms only of the primitives of the nominalist theory, and

vice versa? And if Shapiro’s translations are not conservative extensions, then why should

proponents of either theory grant them any credence in the first place?

What is the upshot of all of this? Corcoran remarks that definitional equivalence,

“preserves categoricity, decidability and completeness” (1980, 231).12 None of these notions

are explicit factors concerning the ontological commitments of a theory (at least when the

theory in question is first-order). Thus, if two theories are definitionally equivalent in

Corcoran’s sense, they need not have identical ontological commitments.

Perhaps the comparison with Corcoran is misleading; perhaps Shapiro is not attempting

to capture a pre-existing notion of definitional equivalence but is instead attaching a new

sense to the term.13 Granting Shapiro’s usage, Chihara provides a prima facie example

of two theories that appear to meet Shapiro’s conditions for definitional equivalence (or,

“Shapiro equivalence” in Chihara’s terminology), but clearly have distinct commitments:

Suppose theory A is a first-order theory about Mr. Jones’s cats and theory Bis a first-order theory about Mr. Smith’s dogs. And suppose that A and Bare Shapiro equivalent (the domain of A consists of the cats of Jones, and the

12For a similar result, consult (Pinter 1987).13Charles Chihara writes that, “it should be noted that some logicians have questioned the appropriateness

of Shapiro’s conditions for definitional equivalence” (2004, 186).



domain of B consists of the dogs of Smith; what A says about the numberof males Jones has, B says about the number of males Smith has—and itjust happens to be the case that both theories are true). Theory A is no merenotational variant of theory B, for B is a theory about dogs, not cats. One couldnot explain the barking emanating from Smith’s house by appealing to theoryA. (ibid., 188)

Although I am sympathetic with this response, I imagine Shapiro would not be much

troubled by it. On the epistemic front, I suppose one could bite the bullet and accept

theory A as providing a good explanation for the barking emanating from Smith’s house.

Suppose that ‘Some dog barks’ is provable in B. If A and B are definitionally/Shapiro

equivalent, there must be a translation of this theorem into A, say, that ‘Some cat meows,’

that is provable in A. If one is is a position to know that A and B are related in this way,

then I see no reason why ‘Some cat meows’ fails to constitute evidence for ‘Some dog

barks.’ As an explanation for ‘Some dog barks,’ it might not be accorded the same virtues

as an explanation solely in terms of B, but that does not imply that the explanation in

terms of A is altogether useless. Shapiro endorses a similar reply regarding mathematical

theories:

The neutral observer sees that if the realist finds good reason to believe Φ,according to his own lights, then the [nominalist] can find good reason tobelieve Φ′ in her framework. The neutral observer sees that the fictionalist hasa good reason to believe Ψ if and only if the realist has a good reason to believeΨ′. (Shapiro 1997, 225)

The damaging suggestion, however, is that in Chihara’s example, the “Shapiro equiva-

lence” between A and B implies that Mr. Jones’s cats are identical to Mr. Smith’s dogs

(and on the ideological front, that doghood is indistinct from cathood). I suspect Shapiro

would maintain that this conclusion misses the point—what is of interest are the onto-

logical/ideological commitments of mathematical theories, and not of theories in general.

Indeed, Shapiro gives no indication that definitional equivalence has ontological implica-

tions for theories in general; his result holds only “as far as mathematics goes” (ibid., 238).

A truer test would involve concocting a mathematical example of definitionally equivalent



theories that nevertheless have distinct ontological commitments.

Chihara invites the reader to compare Russell’s simple theory of types with the standard

set-theoretic version of type theory (2004, 188-91). According to Chihara, there are relatively

smooth translations between the language of Russell’s simple theory of types and the

set-theoretic version of type theory. However, he points out that Zermelo’s Well Ordering

Theorem is not, strictly speaking, a theorem of Russell’s simple theory of types. In

Russell’s type theory, a proof of well-ordering requires the conditional assumption of the

Axiom of Choice. Meanwhile, the set-theoretic version of type theory assumes Choice

as an axiom, and subsequently includes well-ordering as a theorem. It follows that inter-

translatable theories, even in mathematics, do not preserve theoremhood. However, on my

understanding of Shapiro, the preservation of theoremhood is a condition for definitional

equivalence, and not a consequence of it. Thus, one could dispute whether Chihara’s

mathematical example poses a genuine problem for Shapiro.14 Nevertheless Chihara

claims that the moral is that,

All that [Shapiro] has done is to provide us with a method for translating sen-tences of the Constructibility Theory into sentences of simple type theory and amethod for translating sentences of simple type theory into sentences of theConstructibility Theory, these translations preserving certain mathematicallysignificant relationships. . . We can have such methods of translation, even whena sentence in one theory is quite different in meaning and logical significancefrom the sentence of the other theory into which it is translated. (2004, 192)

Shapiro does not claim that his translations preserve meaning (Shapiro 1997, 224),15 and

one would search in vain for any passage in which he denies or appears to disagree with

any statement in the previous quotation. Nevertheless, the ontological commitments of a

theory are typically exposed by interpreting the theory, i.e., by providing a semantics for the

theory. Thus ontological commitment is, ordinarily, linked to meaning. The thought that

Constructibility Theory and simple type theory have distinct ontological commitments is

14It would be revealing if, e.g, Shapiro somewhere argued that Russell’s simple theory of types exhibitedthe same structure as the set-theoretic version of type theory, but to my knowledge he has never done this.

15In the next section I explore what moves are available if this claim is dropped.



bound up with Chihara’s contention that sentences of simple type theory are distinct in

meaning from their correlates in Constructibility Theory.

This discussion raises an important distinction between the ontological/ideological

commitments held by a theory in virtue of its quantifiers/predicates, and the ontologi-

cal/ideological commitments held by a theory in virtue of its structure. The traditional

Quinean ontological project exposes commitments solely in virtue of quantifiers (or, to

be more specific, solely in virtue of first-order quantifiers in appropriately regimented

languages). But Shapiro is not a typical ontologist when it comes to mathematics: Two

theories are said to have the same mathematical ontology when they are definitionally

equivalent. The import here is not that, despite appearances to the contrary, the theories

are quantifier-committed to the same ontology. Rather, the point is that, in virtue of being

definitionally equivalent, each theory describes the same structure. Subsequently both

theories share in any commitments to be had in virtue of their exemplifying the same

structure. And, prima facie, two theories can share in such structure-commitments while

having distinct quantifier-commitments.

Thus the “certain mathematically significant relationships”—i.e., the structural properties

of mathematical theories—are what inflate the mathematical realm. Two definitionally

equivalent theories are said to have the same ontological commitments, as far as math-

ematics goes, because of the commitments they have in virtue of exemplifying the same

structure. Moreover, these are kinds of relationships that Chihara seems happy to recog-

nize. Therefore, it again seems possible for Shapiro to accuse Chihara of missing the point.

Mathematical ontology is not determined by the semantics of mathematical theories but

instead by the structural properties of mathematical theories. But this claim is as bold as it

is unobvious. What is Shapiro’s motivation for contending that the ontology of mathemat-

ics is highlighted by the structure of mathematical theories, when for most other kinds of

theories this is not thought to be the case? And is this contention plausible on independent

grounds? In a short while I shall consider in detail what the mathematical structuralist


Shapiro’s Challenge 92 The Paraphrase Response

position is and how mathematical structuralism relates to mathematical ontology, but first

I would like to examine another possible reply that might be made on Shapiro’s behalf.

Those not wishing to lose the thread of the main dialectic may proceed directly to §6

without loss.

2.4 The Paraphrase Response

Modal nominalist accounts of mathematics are often viewed as strategies for paraphrasing

ordinary mathematical assertions into ontologically innocuous assertions. John Burgess,

Bob Hale, and Gideon Rosen have all pursued objections to nominalism (including modal

nominalism) on the grounds that paraphrase is incapable of paring down ontological

commitments. Something like this view of modal nominalism is suggested by Shapiro’s

insistence that it is possible to translate sentences of set theory into sentences of modal

nominalist theories, and vice versa. I would like to consider whether one of Shapiro’s

conclusions—that modal nominalist accounts of mathematics fail to eschew commitment

to mathematical objects—can be better supported by viewing modal nominalist accounts

of mathematics as paraphrases of ordinary mathematical assertions. The reader should

be aware that Shapiro nowhere endorses this tactic, and he openly acknowledges that

the translations needed to establish definitional equivalence are not meaning-preserving

(ibid.). I argue that these objections ultimately fail because modal nominalist accounts of

mathematics are not mere paraphrases of ordinary mathematical languages.

What important insight is available if modal nominalist accounts of mathematics are

viewed as paraphrases of ordinary mathematical assertions? Intuitively, if a sentence S ′

is a paraphrase of a sentence S, then S and S ′ can both be used to express the very same

proposition; S and S ′ make the same claim about the world. And if S and S ′ make the

same claim about the world, then it is hard to imagine how S and S ′ could differ in their

commitments. On the other hand, if S and S ′ differ in their commitments, then intuitively

they do not make the same claim about the world. It is subsequently difficult to see why



S ′ should be regarded as an acceptable paraphrase of S. In the present case, either modal

nominalist accounts of mathematics are acceptable paraphrases and so do not differ in

their commitments from platonism, or else modal nominalists accounts of mathematics

reduce commitments and so fail as paraphrases.

The dilemma of the previous paragraph was posed quite some time ago by William

Alston. Alston asks the reader to compare pairs of sentences like the following:

1. There is a possibility that James will come.

2. The statement that James will come is not certainly false.

On the surface, (1) appears to be committed to the existence of things called ‘possibilities,’

whereas (2) appears to have no such commitment. However, both assertions appear to the

untutored eye to be identical in meaning, and those who are loathe to admit possibilities

into their ontology are likely to regard (2) as an acceptable and ontologically innocuous

rendering of what one says when uttering (1). In response to this Alston remarks,

Now it is puzzling to me that anyone should claim that these translations “showthat we need not assert the existence of” possibilities. . . “in communicatingwhat we wanted to communicate.” For if the translation of (1) into (2), forexample, is adequate, then they are normally used to make the same assertion.In uttering (2) we would be making the same assertion as we would make if weuttered (1), i.e., the assertion that there is a possibility (committing ourselves tothe existence of a possibility) just as much by using (2) as by using (1). If, onthe other hand, the translation is not adequate, it has not been shown that wecan, by uttering (2), communicate what we wanted to communicate when weuttered (1). Hence the point of the translation cannot be put in terms of someassertion or commitment from which it saves us. (1958, 10)16

The ontological reductionist faces a dilemma: Either accept that the translations preserve

the statement being expressed, and so preserve ontological commitments, or admit that

the translations eschew ontological commitments but subsequently fail to preserve the

statement expressed. What is it that determines whether a sentence is ontologically

committed to possibilities?16I have made inessential changes in numbering.



. . . whether a man admits (asserts) the existence of possibilities depends onwhat statement he makes, not on what sentence he uses to make that statement.One admits that possibilities exist whenever he assertorially utters (1), or anyother sentence which means the same. . . It is a question of what he says, not of howhe says it. (ibid., 13)

Thus, Alston locates ontological commitment in the propositional contents expressed

by sentences. Since (1) and (2) intuitively both express the same proposition, they have

identical commitments. Alston goes on to suggest that the value of such translations

consists in their psychological effects:

It is the seductive grammatical family likeness of sentences like (1) which renderthem objectionable, not any assertion of the existence of possibilities they carrywith them, in any intelligible sense of that term. (ibid., 16)

Compare (1) with the following:

3. There is a beer that Larry will drink tonight.

Both (1) and (3) share the grammatical structure ‘There is a/an x that F s.’ It is perfectly

acceptable to ask, about the beer mentioned in (3), where it is located in space and time,

whether it is an ale or a lager, when it was brewed and bottled, etc. Experience with

(3) suggests that sentences with a similar structure will also admit as intelligible similar

kinds of questions about the objects denoted by terms that occupy the same position in

the sentences. Upon returning to (1) it is found to be rather nonsensical to ask, about

the possibility that James will come, where this possibility is located in space and time,

whether it is an ale or a lager, etc. Such questions lack clear answers and lead to skepticism

about the existence of possibilities. Alston claims that the translation from (1) to (2) frees

one from having to regard such questions as well-formed, and so alleviates much of the

mystery surrounding entities like possibilities.

I am not inclined to comment on Alston’s hypothesis about the substratal value of

translation, however I do not see how entities like possibilities are any less mysterious

just because certain awkward questions about them are evaded—if anything, the fact


Shapiro’s Challenge 95 Nominalist Paraphrase as Synonymy

that ordinary kinds of questions need not be asked about them make possibilities more

mysterious. In any case, Alston’s discussion is predicated on a skepticism about reductive

paraphrase. A similar position is advanced by Frank Jackson:

In short, the crucial question is not what one assents to in the object language,but what one assents to in the metalanguage which explicitly states the seman-tical roles of the terms in the object language. (1980, 310)

For Jackson, a metalanguage is ontologically committed to Ks, “when it entails ‘that there

are things which “is K” applies to’ ” (ibid., 311). Although the details are distinct, I take

Jackson to be making essentially the same point as Alston, for I understand the function

of the metalanguage, on Jackson’s account, to be exposing “what it is we are trying to

communicate” in uttering a sentence in the object language. Cindy Stern also makes

essentially the same point:

. . . answers to ontological questions cannot be found by looking at represen-tations of statements in a formal language we take to be logically perspicu-ous. . . what is needed is to answer the question of what the truth conditions arefor the relevant statements, regardless of how we represent them. (1989, 42)17

The success of these general remarks as applied to the case of modal nominalism

depends on the nature of the relationship between ordinary mathematical assertions

and their modal nominalist counterparts. The proposed case against modal nominalism

envisions modal nominalist accounts of mathematics as synonymous with or analytically

equivalent to ordinary mathematical assertions.

2.4.1 Nominalist Paraphrase as Synonymy

If modal nominalist accounts of mathematics are taken to be synonymous with ordinary

mathematical assertions, then Alston’s result applies directly: Assertions in modal nomi-

nalist theories mean the same things as ordinary mathematical assertions, and so modal17It should be noted that Stern does not follow Alston in completely abandoning the idea that paraphrase is

incapable of paring down ontological commitments; her concern is more to show that providing paraphrasescannot ever constitute a positive reason for eschewing commitments. According to Stern, paraphrase is adefensive measure to be supplemented by ulterior reasons for being suspicious about the entities in question(ibid, 36).



nominalist theories do not eschew commitment to mathematical objects. Arguments that

fit this mold have been made by John Burgess, Bob Hale, and Gideon Rosen.

Burgess begins by assuming that the acceptance of a statement such as,

4. There are numbers that are greater than 101010 and are prime.

clearly entails accepting statements such as,

5. There are (such things as) numbers (Burgess 2001, 429).

However, one cannot accept (5) without being ontologically committed to numbers. Ac-

cording to Burgess, nominalists seek to avoid the commitments implicit in (5) by adopting

a paraphrase of (4). He offers a possible nominalist paraphrase of (4):

6. There could be constructed numerals that were greater than 101010 and were prime.

which entails only that,

7. There could be constructed (such things as) numerals (ibid., 431).

Against nominalism, Burgess argues that,

If the argument of. . . nominalists is supposed to be that they are warranted(despite their professed nominalism) in asserting (4) because its paraphrase (6)is assertable, parity of reasoning would suggest that one is warranted also inasserting (5), since its paraphrase (7) is assertable (ibid., 437).18

Burgess targets this argument at a kind of view he elsewhere describes as hermeneutic

nominalism.19 The hermeneutic nominalist alleges that her account of mathematics pro-

vides an attractive interpretation of ordinary mathematical languages; she claims to be

giving an account of what mathematical assertions really mean. In this sense the nominalist

is providing an account of why it is acceptable to assert statements such as (4) and (5),

because all these statements really mean can be captured by asserting statements like (6)

18I have made inessential changes in numbering.19See chapter four for more details on Burgess’s arguments against nominalism.



and (7). Burgess, like Alston before him, maintains that the tables can be turned and that

the paraphrases (6) and (7) make only an illusory gesture at ontological reduction.

Bob Hale presses a very similar point while discussing Hellman’s method for providing

a modal-structural interpretation of mathematical theories. Recall that Hellman posits two

kinds of modal claims: First are the necessary conditionals of the form AX → p where p is

a consequence of axioms AX . And second, to avoid vacuity, are claims stating the possible

existence of models of the axioms AX . These latter claims are known as “Modal Existence”

claims. Hale writes,

It is a necessary condition for its adequacy that the scheme takes ordinaryarithmetic truths into truths and arithmetic falsehoods into falsehoods. Sincethis condition is certainly not met unless the Modal Existence claim is true,reason to think the scheme adequate must include reason to think that claimtrue. (1996, 133)

Hale’s official target in writing this is Hellman’s justification for supposing that the modal

existence claims are true. Extending Hale’s remarks somewhat, any thought that the modal

existence claims are true, via the translation scheme, is just as much a thought that the

relevant mathematical assertions are true. Otherwise, the scheme might take arithmetic

falsehoods into truths, or falsehoods into arithmetic truths, etc. Any reason to be confident

that the modal existence claims are true requires believing that the translation scheme is

adequate, which entails believing that certain mathematical claims are true.

The argument I extract from Hale can be dealt with rather quickly. What is preserved

in the translation scheme between ordinary mathematical assertions and modal-structural

assertions is not truth simpliciter but theoremhood. Reason for thinking the scheme adequate

need not involve assuming that the relevant mathematical assertions are true, but instead

that they are truly theorems of the mathematical theory in question. The theory itself need

not be regarded as true.

A similar response is available for the modal nominalist concerned by Burgess’s ar-

guments. Let me first say that I am unaware of any nominalist (modal or otherwise)



who endorses statements such as (6) and (7). Moreover, (6) seems to embody a category

mistake—numerals are not the kinds of things that can have properties like being prime, or

being greater than 101010 .20 Nevertheless modal nominalist theories are bound to include

pairs of assertions, (6i) and (7j), that occupy the same mathematical roles as (4) and (5)

do in ordinary mathematics. The question raised here is just what the modal nominalist

paraphrases are lauded as preserving. If modal nominalist paraphrases are understood as

synonymous with ordinary mathematical assertions, then they ought to preserve meaning

in both directions. The problem with understanding modal nominalist paraphrases in this

way is that no modal nominalist that I am aware of has ever described their account of

mathematics as a meaning-preserving paraphrase of mathematical language. Nor am I

aware of any modal nominalist who maintains that her account of mathematics is designed

to uncover the true meaning of mathematical assertions. Rather, the modal nominalist

views I defend in this dissertation are interested in answering modal questions about

whether it is possible to carry out various kinds of scientific and mathematical reasoning

without quantifying over mathematical objects. And even if modal nominalists were in

the business of attempting to tell the world what mathematics is really about, there is

no reason to suppose that they would accept the inference from (4) to (5). If all (4) really

means is what is said in (6i), then (7j) does not imply the truth of (5) in any ontologically

significant way.

Gideon Rosen presents a similar case against nominalism. Using a thought experiment

dating back to Frege, Rosen asks the reader to compare two rather simple geometric

languages, L and L∗. The language L contains vocabulary for attributing properties

to lines and also includes the two-place relation x‖y that is to read ‘x is parallel to y.’

Intuitively, to say that line a is parallel to line b is to say that a and b have the same direction.

However, L does not contain any terms that refer to any such things as directions. L∗ is an

extension of L which is to contain a new predicate d(x) that is to read ‘the direction of x,’

20Thanks to Susan Vineberg for this observation.



which is a referring phrase that not only refers to a line x but also to an object that is the

direction of the line x. It seems obviously true that the following two statements mean the

same thing:

8. a‖b,

9. d(a)=d(b),

and, that subsequently, the move from L to L∗ does not importantly distort assertions

about lines. However, if Frege’s analysis of language is granted, then a statement like (9)

is committed to the existence of directions, because ‘the direction of. . . ’ functions in the

statement as a singular term. So (9) is only meaningful and true if there exist things like

directions. But (8) and (9) mean the same thing; they are analytically equivalent. According

to Rosen,

The fact that such a move was always possible shows that a commitment toabstract objects was in a sense already implicit in the original language. True, Lcontains no devices for referring explicitly to abstract objects. Still, what we’vejust seen is that the obvious truths of L express propositions that are true onlyif abstract objects exist, in the sense that they are analytically equivalent tostatements which refer explicitly to DIRECTIONS. (1993, 167)

Thus, the nominalist who asserts both (8) and the analytical equivalence of (8) and (9) and

who, “denies the existence of DIRECTIONS implicitly contradicts himself whenever he

so much as asserts that two lines are parallel” (ibid).

If pressed, I would reject the notion that ‘the direction of. . . ’ functions as a genuine

referring term, but that is a discussion for another occasion. My response to Rosen is

similar to my response to Burgess and Hale: The kinds of modal nominalist views I

defend in this dissertation are not in the business of providing analytically equivalent

expressions of platonistically construed mathematical propositions. Neither Field nor

Chihara nor Hellman claim that their accounts of pure mathematics involve assertions that

are identical in meaning to mathematical assertions interpreted platonistically. Again, the

purpose of modal nominalist reinterpretation and reconstruction is primarily to show that


Shapiro’s Challenge 100 Reply to the Paraphrase Response

it is possible to account for the obtaining of mathematical truths and the applicability of

mathematics to science21 without invoking platonistic ideas about mathematical truth and

scientific applications. What is under dispute is the presumption often made by platonists

that mathematics comes with a platonistic interpretation directly from the mathematicians.

Rosen’s Fregean case against nominalism is guilty of making this presumption, and

therefore does not address the views about which I am concerned.

2.5 Reply to the Paraphrase Response

My reply to the paraphrase response, then, is rather curt: It is an uncharitable distortion

to view modal nominalist accounts of mathematics as mere paraphrases of ordinary

mathematical assertions; paraphrase is not the vehicle through which modal nominalists

eschew commitment to mathematical objects. Thus the arguments of the previous section

do not touch any of the modal nominalist views I defend in this dissertation.

Nevertheless it is not clear to me that the arguments of the previous section are ulti-

mately effective against a paraphrase-style nominalism (modal or otherwise). It seems to

me that these arguments turn on two important presuppositions: that mathematical asser-

tions are true simpliciter and that these assertions are ontologically committing. But surely

a paraphrase-style nominalist would be interested in challenging both presuppositions.

If, in opposition to the view that mathematical assertions are true simpliciter, it is more

appropriate to view mathematical assertions as, e.g., truly theorems or consequences of the

axioms of mathematical theories, then nominalist paraphrase can proceed by preserving

theoremhood (as opposed to truth simpliciter). And if one does not presuppose that the

face-value or literalist interpretation of mathematical language is ontologically committing,

then it cannot be complained that nominalist paraphrases do not eschew commitment

to mathematical objects, since this commitment is not there to begin with.22 I happen to

21In Field’s case only the second of these activities is attempted.22Of course, a nominalist who did not already believe that mathematical language was not ontologically

committing would probably not see the need to produce nominalist paraphrases (except perhaps to convinceher platonist foes that such commitment is not in fact present).



believe that both presuppositions should be challenged. I argue elsewhere in the disserta-

tion23 that, on naturalist grounds, it is no presupposition of mathematics that mathematical

languages are ontologically committing. In this section I would like to offer some broadly

naturalistic reasons against the presupposition that mathematical theories and assertions

should be regarded as true simpliciter.24 The upshot is that, from a naturalistic perspec-

tive, neither presupposition is legitimate, and hence that paraphrase-style nominalism

constitutes a coherent position from which to eschew commitment to mathematical objects.

As I understand matters, the arbiters in the nominalism/platonism dispute are the

implications of professional mathematical discourse, and not the implications of ordinary

mathematical assertions as understood by lay individuals. The simple fact is that the

mathematician’s understanding of basic arithmetical assertions is not the same as the

layperson’s understanding of mathematical assertions. There is no compelling reason to

believe that both kinds of assertions should be treated as semantically on a par. For the

mathematician, “2+2=4” is a rather uninteresting theorem of arithmetic. It is not the kind of

assertion made in a vacuum. For the layperson, “2+2=4” is a rather simple assertion often

made in isolation. The same goes for epistemology. How the lay person comes to acquire

justified beliefs about mathematics is altogether different from how the mathematician

does this. And there is no guarantee that they arrive at what are recognizably the same

beliefs. For the mathematician comes to believe that “ ‘2+2=4’ is a theorem of arithmetic,”

whereas the layperson comes to believe just that “2+2=4.” Even supposing that it is

acceptable to assume that lay persons actually know that statements like “2+2=4” are true

in a mathematically perspicuous way, there is a problem of accounting for what justifies

these beliefs. If “2+2=4” makes a true statement about abstract mathematical objects, then

any account of how one comes to believe that “2+2=4” must also specify how it is one

comes to know that abstract objects exist—there are no epistemological free lunches. I take

it that “intuitive perception of a realm of abstract objects” is an unacceptable response here,

23Chapter four, §5.24Cf. my discussion of Penelope Maddy’s related position, Arealism, in chapter five, §4.



but most especially if offered on behalf of lay persons. The problem is, how humans actually

form beliefs about statements like “2+2=4” seems to have nothing whatsoever to do with

abstract objects. Why should belief in “2+2=4” compel anyone to believe that there exist

abstract objects?

Frege’s response (and the response of those who follow him) is that “2+2=4” is an

obvious truth, indeed even a logical truth, and that the platonist’s semantics provides the

best account of what the statement means. But what explains the obviousness of “2+2=4”?

The best explanation, I submit, must not posit any mysterious powers either on the part

of mathematicians or on the part of laypersons (and so again, intuitive perception of a

realm of abstract objects is out of the question). What evidence do lay people have for

believing that 2+2=4? I suppose their evidence is primarily authoritative, as is much of their

evidence concerning the truths of mathematics. Where their evidence is not authoritative

it is experiential. Whenever they have collected two objects with another two objects they

have always (or for the most part) found there to be four objects all together. It is prima facie

implausible that either source of warrant justifies the layperson in believing that abstract

mathematical objects exist.

What evidence do the “authorities” have? Likely the fact that it is a theorem of

arithmetic that “2+2=4.” But now the question becomes, is arithmetic a theory whose truth

can be known through something resembling a priori insight alone? The answer is a firm no.

Mathematical theories are constructed with some intuitions in mind—the numbers ought

to have properties x, y, z, and so the axioms are designed to accommodate this. But there

is no guarantee that these axioms are free from contradiction, as evidenced by the early

stages of set theory. And often enough, intuition is too poor a guide for pronouncing on

the nature and properties of the mathematical objects in question—again, the development

of set theory is evidence of this—for what, if anything, does the notion of ‘set’ amount to?

Just what extension of ZFC falls under the concept ‘set’?

Any argument for the existence of mathematical objects that proceeds from citing



supposed obvious, apparent, or commonsense truths of mathematics fails to place mathe-

matical assertions in their proper context. Mathematically, assertions such as “2+2=4” are

consequences of mathematical theories. And no amount of reflection on the obviousness

of truths such as “2+2=4” will ever deductively entail the truth of any interesting mathe-

matical theory. One reason for this is that many of the obvious or commonsense truths

of arithmetic are also logical truths of first-order logic with identity—detracting from the

idea that Fregean semantics provides the best explanation for the indubitability of the basic

truths of arithmetic.25 Nevertheless I do not deny that some notion of self-evidence plays a

role in the formulation of mathematical theories (and hence, in the generation of mathe-

matical knowledge). If a theory is finitely axiomatizable then it is perhaps possible that one

could regard each of its axioms as self-evident. However, most first-order mathematical

theories include axiom schemas and so are not finitely axiomatizable from an epistemic

vantage point—for instance, the first-order Peano axioms include an induction schema.26

What is the evidence showing that humans are justified in believing that axiom schema are

true simpliciter? Is it possible to avoid appealing to mysterious “mathematical intuitions”

(the likes of which Kurt Godel championed) in justifying beliefs in axiom schema? Can it

even be guaranteed that axiom schema are ultimately justified on a priori grounds rather

than on pragmatic grounds?

I am suggesting that self-evidence has its limits, especially in mathematics. For instance,

Penelope Maddy presents a case study of the Axiom of Choice.27 Her findings indicate

that Choice became accepted, not because mathematicians found it to be self-evident,

but instead because of the fruitful consequences of adopting the axiom. If that is correct,

then mathematical theories need not, and in many cases, are not guided by self-evidence

25For a more detailed argument to this effect, see (Leng 2010, 90-4). Frege, of course, thought arithmeticwas reducible to self-evident logical principles, but he was infamously wrong about that—Frege’s system, asis well known, is inconsistent.

26Second-order PA is finitely axiomatizable, but its induction axiom includes universal quantification overproperties. I understand there to be negligible epistemic gap between confidence in the first-order inductionschema and confidence in the second-order induction axiom.

27See (Maddy 1997) and (Maddy 2011).



alone, but also by considerations of mathematical fruitfulness. But then mathematical

theories cannot be justified wholly (or perhaps at all) by self-evident principles—some sort

of “mathematical experience” is inevitably necessary.28

I say these things to suggest that mathematical assertions—even the existential ones—

must be understood in their proper context. And the proper context for making mathematical

assertions is within the grips of mathematical theories. If it is not granted that mathemati-

cal theories are true simpliciter, then paraphrases can accomplish their goals while only

preserving theoremhood. The platonist needs to provide persuasive reasons either for

regarding mathematical theories as true simpliciter, or for regarding assertions like “2+2=4”

as statements that are about mathematical objects, but that nevertheless are made true in

isolation from their status as theorems of mathematical theories. A paraphrase-style nomi-

nalist should express interest only in the ontological commitments of mathematical theories,

and not the ontological commitments of miscellaneous mathematical-looking assertions.

Thus the best scenario for the platonist involves showing that nominalist paraphrases are

paraphrases of mathematical theories that are themselves true simplicter. But then a ques-

tion is raised about the source of warrant for regarding such theories are true simpliciter.

What I have called into question is the idea that platonists are entitled to be confident that

mathematical theories are true simpliciter simply because some mathematical assertions

appear to be obvious or self-evident. I admit that this looks like unhelpful burden-shifting,

and it very well may be.29 Nevertheless there appears to be space for a successful pro-

gram of nominalist paraphrase, if it can be shown that the platonist is not straightaway

entitled to herald mathematical theories as true simpliciter (and if, as I argue in chapter

28For more discussion, see (Shapiro 2009). Shapiro argues that even Frege appealed to considerations ofwhat Maddy would call “mathematical fruitfulness” in justifying his Basic Law V.

29Should one invoke Quine’s notion that ontological commitment be tied to the “best overall theory,” thensimplicity considerations can be mustered in support of nominalist theories, provided that one distinguishesbetween two senses of “best overall theory”: (1) The best overall theory as the “true” theory, and (2) the bestoverall theory as the theory that is most propitious in practice. As far as I can tell, the nominalist (includingthe paraphrase-nominalist) claims only that her theories are better in the first sense, and that the fitnessof theories in the second sense is irrelevant when it comes to matters of ontology. My sense is that thearguments I have entertained against the paraphrase-nominalist are not Quinean in spirit.


Shapiro’s Challenge 105 The Structuralist Response

four, that platonists are not entitled to the presupposition that mathematical language is

ontologically committing). Still, it is worth reminding the reader that neither Chihara nor

Hellman nor Field are interested in providing mere paraphrases of ordinary mathematical

assertions, and it is for this reason that the Paraphrase Response ultimately fails as a way

of bolstering Shapiro’s objections to modal nominalism. Of course, Shapiro himself is

likely to bolster his objections in a way that relies on his structuralist understanding of

mathematics, and it is to Shapiro’s structuralism that I now turn.

2.6 The Structuralist Response

Mathematical structuralists are inspired by the slogan that “mathematics is the science

of structure up to isomorphism” or some other similar rallying call. Structuralists find

significance in the fact that, as far as mathematics goes, mathematicians are more interested

in constructing and examining various kinds of relationships between mathematical objects

than in studying mathematical objects themselves. That is to say that mathematicians are

most interested in the structural properties described by mathematical theories. This view

raises a number of questions that include, but are certainly not limited to, the following:

What is a mathematical structure? Is it a kind of mathematical object? And if so, what is a

structure like? Do there exist any interesting relationships that hold between structures, say,

between the natural number structure and the real number structure? Do structures exist

over and above ordinary mathematical objects like numbers? Or does a structure depend

for its existence on a collection of objects that stand in the appropriate kinds of relations

so as to constitute a structure of that kind? Why suppose that structures exist in the first

place?

I cannot address all of these questions in detail here; mathematical structuralism is

much discussed in the literature. In particular, the question regarding whether there exist

any interesting relationships between mathematical structures has led to a great deal of

discussion. For instance, it is rather intuitive to suppose that the natural number 2 is


Shapiro’s Challenge 106 Resnik and Patterns

identical to the real number 2. However, 2 has properties qua natural number that it lacks

qua real number, and vice versa. If the only mathematical significance of the number

2 lies in the relations in which it stands to other members of the natural/real number

structures, then by the non-identity of discernibles, the 2 of the naturals is not the same as

the 2 of the reals. Structuralists usually respond that in mathematics, ontology is always

relative to structure, and that identity conditions for mathematical objects cannot be given

in generality. The debate rages on.30 For better or worse I am not much interested in

this debate. My interests instead lie in coming to understand just what sorts of things

mathematical structures are and whether there exists an account of structure that is capable

of validating Shapiro’s criticism that modal nominalists are committed to mathematical

structures.

The two major structuralist views to be examined are Shapiro’s ante rem structuralism

and Michael Resnik’s structuralism. Each hopes to shed light on the epistemology of

mathematics by constructing a philosophy of mathematics that makes important use of the

human ability to recognize patterns. Since pattern recognition is allegedly a naturalistically

kosher faculty, if mathematical knowledge is ultimately knowledge of patterns, then

perhaps a structuralist version of platonism can solve the epistemological difficulties

facing the traditional, “objects” platonist views. I begin with Resnik’s account of structures

before moving on to Shapiro’s. The reader is again reminded that my primary interest

is in plumbing these views for a conception of structure (and the relationship between

structure and ontology) that validates Shapiro’s objections to modal nominalism; what

follows is not offered as a comprehensive discussion of mathematical structuralism.

2.6.1 Resnik and Patterns

According to Resnik, humans possess the capacity to recognize all kinds of patterns in

the natural world. It is an uncontroversial datum that humans form more or less justified

beliefs on the basis of this capacity. It is through pattern recognition that I judge my

30For discussion, see (Hellman 2005) and (MacBride 2005), and the references therein.



television screen to be similarly proportioned to the screen of my smart phone, that I

recognize the fourth word of this clause to be identical to the fourteenth, etc. Resnik

proposes to include mathematics in the canon of what can be known through the faculty

of pattern recognition. For, according to Resnik, mathematical theories exhibit patterns—

mathematical patterns—and these “are known in the same way as, say, linguistic or

musical patterns. . . by putting mathematics in the same epistemological context as music

or language we remove some of the mystery enveloping platonism” (1975, 34).

But what is a pattern? Resnik says that a pattern31 consists of one or more objects—the

positions of the patterns—that stand in certain relationships (1997, 202-3). And, as far as

a pattern is concerned, each position is distinguished only through the relationships it

bears to the other positions of the pattern. A favorite example of Resnik’s is a triangle

ABC. Relative to this triangle one can identify the points that serve as the positions for its

vertices, but A, B, and C, when considered independently of any reference to the triangle,

“are indistinguishable from each other and any other points” (ibid., 203).

Another example is the natural numbers under the successor relation (N, S). The

positions of this pattern are the numbers, and the numbers are distinguished only through

the relationships they bear to one another. For instance, all that it is to be the number zero

is to be the position that is not in the range of the successor relation. All that it is to be the

number one is to be the successor of zero. All that it is to be the number two is to be the

successor of one, etc. Provided that there exist enough of them, any objects whatsoever

can fill the positions of the natural number structure. Other mathematical patterns include

the real numbers under the less-than relation (R, <) and the cumulative hierarchy under

epsilon (V, ε). For Resnik, N, R, and V are not antecedently given items of mathematical

ontology. Natural numbers, real numbers, and sets do not have identity criteria on their

own; their identification is always relative to fixing on a particular pattern (Resnik 1981,

545).31Resnik prefers to use the term ‘pattern’ where Shapiro would use the term ‘structure’; nevertheless

Resnik uses these terms interchangeably, and I follow his lead.



Although Resnik demurs from constructing a formal theory of patterns,32 he does

articulate several relations that hold between them.33 The most fundamental relationship

between patterns is occurrence. Occurrence is a transitive and reflexive relation that holds

between two patterns; P is said to occur within Q when there is a third pattern R such

that P is isomorphic to R, any position of R is a position of Q, and all relations of R are

definable in Q. For example, the natural numbers under successor (N,S) occur within the

natural numbers under less-than (N,<) because the successor relation is definable relative

to the less-than relation.34 A special case of occurrence is the sub-pattern relation. P is a

sub-pattern of Q if and only if P occurs within Q and every position of P is a position of Q.

Finally, two patterns are said to be equivalent or “essentially the same” under the following

conditions:

. . . let us call a pattern P a truncation of a pattern Q if every position and relationof P is also one of Q. Then a pattern P will be said to be equivalent to a patternQ just in case there is a pattern R which is a sub-pattern of both P and Q and ofwhich both P and Q are respective truncations. (Resnik 1997, 209)

(N,S) and (N,<) are both truncations of (N,S,<), and the positions of (N,S,<) are the

positions of (N,S) and (N,<). Hence (N,S) and (N,<) are equivalent or “essentially the

same” patterns.

It turns out that Resnik has quite a bit to say about when a particular pattern exists and

about when one is justified in believing that a theory of a pattern is true. In an early paper,

he writes:32The reason for this is that Resnik is not officially a realist about patterns. According to him, mathematical

theories do not quantify over patterns, and so accepting the existence of mathematical patterns is a distortionof practice (Resnik 1997, 211). However, it is quite common for Resnik to speak as though he is a realist aboutpatterns, and in this section I will treat him as such. Inevitably this discussion will involve some distortionof Resnik’s actual position, but I hope the reader will agree that the perturbation is not specious—especiallygiven the passages to come.

33What follows is drawn from (ibid., 205-9).34In English, x is the successor of y means that y is less than x and there is no z such that y is less than z

and z is less than x. Now the ordinary notion of definability is bi-directional. However, the less-than relationcannot be defined relative to the successor relation in first-order languages. Thus Resnik must appeal toset-theoretic or second-order definability to get the result that < and S are definable relative to one another.See (ibid., 207).



When we “construct” a pattern we will formulate some tentative beliefs aboutit. Hence our conception of the pattern and our beliefs about it can be subjectedto a deductive study during which we attempt to formulate our conception asprecisely as we can and examine its logical consequences. This investigationwill indicate the degree to which our conception is internally coherent andhow well it accords with established mathematics. If it turns out to be highlycoherent and confirmed by our knowledge of the finite patterns from which itarose, then our belief in the existence of the pattern is justified. (Resnik 1975,36-7)

Similar sentiments are expressed in a later articulation of his position:

We go through a series of stages during which we conceptualize our experiencein successively more abstract turns. At the last stage we leave experiencefar enough behind that our theories are best construed as theories of abstractentities. (Resnik 1982, 99)

. . . a pure theory of a pattern is justified to the degree to which we have evidencefor its consistency and categoricity, with the former having a higher priority.(ibid., 101)

For example, early on in their lives humans form the ability to mentally group or collect

together objects. At some point in time they form the ability to enumerate the members

of these groups, if there are few enough objects in them. Thus they become familiar with

the early stages of the progression of the cardinal numbers: one, two, three,. . . This is still

a concrete activity. Shortly enough, however, comes the recognition that this sequence

of counting numbers can be represented in written language as numerals, and that these

numerals can be used to count or list arbitrary sets of objects. Thus, a process of abstraction

begins. Some may take an interest in the progression itself and begin a more studious

investigation. These individuals soon realize that the progression has no natural ending

place, but that due to physical limitations, they could never envision the progression to

be “completed.” Still, there is an interest in answering the question, “How long does

this progression go on?” Answering this question requires another leap of abstraction,

from numerals to numbers. Soon enough the Dedekind/Peano axioms are proposed as

capturing the important properties of the natural numbers. It is at this point that the



“deductive study” of the pattern of natural numbers begins. No contradiction is found in

the derivations of the Dedekind/Peano axioms, and the various arithmetical consequences

prove to be highly useful for ordinary and scientific purposes. It follows that a belief in the

existence of the natural number pattern (N,S) is justified. Nevertheless,

. . . all this depends upon our having unconditional knowledge of some initialstructures, and with respect to those structures our evidence must be non-deductive and indirect. . . I conjecture that at least our early evidence in favorof a pure theory of an “initial pattern” is furnished by the degree of coherencebetween the theory and our beliefs concerning the experience from which ithas been abstracted. (ibid., 101-2)

In this regard, Resnik notes that simpler theories—theories of patterns with fewer positions

and/or less complicated relations (whatever the word ‘complicated’ means here)—are

more likely to be true (ibid., 102).

Resnik is asking the reader to believe something quite astounding—one is asked to

believe that a pattern exists because its postulation is coherent.35 Surely that is an oversight;

possibility does not imply actuality. But as astounding as this sounds, it is not an oversight.

According to Resnik, “in mathematics for a structure to be possible is for it to be actual”

(1985a, 176). He maintains that, “the sort of reasons I advanced for holding that a given

mathematical structure is possible are also (or almost) reasons for holding that it is actual”

(ibid). What is his evidence for these remarkable claims?

But suppose that we grant that there are abstract structures. Then to justify theexistence of a particular abstract structure it suffices to exhibit a template forthings which instantiate that pattern. Thus to show that a specific pattern forhouses exists, it suffices to exhibit a blueprint for houses of that pattern. . .

These considerations show that once we countenance structures, giving aconsistent description or representation of a particular structure is all we needdo to show that such a structure exists. When it comes to structures, thedistinction between the possible and the actual lapses. (ibid., 178)

It is not immediately clear how these remarks are supposed to resolve any of the mys-35A similar belief is shared by Mark Balaguer in defense of his “plenitudinous platonism” (see the first

chapter for a brief overview). And, as will be seen below, Shapiro ends up in a similar position in defense ofhis “Coherence axiom.”



tery surrounding the claim that for mathematics, possibility implies actuality. Perhaps

Resnik is conveying an analog of existence claims in set theory. One way of showing

that a mathematical object is coherent is to show that it can be constructed in set theory

(or in a consistent extension of set theory). If sets are antecedently postulated items of

mathematical ontology—that is, if the full set-theoretic hierarchy exists—then anything

“constructible” out of sets already exists. On this interpretation, Resnik’s claim amounts

to the hypothesis that if structures are antecedently given objects, then any patterns ab-

stracted from them “already exist” in some sense. But just as it is reasonable to ask, “Why

suppose that sets exist in the first place?” so too it is reasonable to ask, “Why suppose that

patterns or structures exist in the first place?” Resnik’s answer is that, “recognizing them,

at the very least, greatly facilitates our theorizing and is in all probability indispensable to

it” (ibid). He later elaborates, explaining that the, “decision to allow [patterns] to serve as a

part of the evidential basis for further mathematical developments would have to be made

from the more global perspective of the benefits to mathematics and science generally”

(Resnik 1997, 238).

Can Resnik’s account of structure existence help Shapiro? Shapiro’s strategy is to

concoct a variety of structure-equivalence between set theory and modal nominalist

theories, and argue that this equivalence precludes modal nominalists from eschewing

commitment to mathematical objects and prevents them from providing a more tractable

epistemology for mathematics. Resnik provides a relation of pattern-equivalence that itself

is structurally similar to Shapiro’s conception of definitional equivalence. Resnik’s idea is

that two patterns are equivalent when both are truncations of some larger pattern. What

matters here are the positions and relations of the patterns in question. A case can indeed

be made that the modal nominalist theories in question are pattern-equivalent to some

platonist theory. Chihara’s Constructibility Theory is pattern-equivalent to simple type

theory: Chihara’s positions are filled by open-sentence tokens, whereas the positions of

type-theory are propositional functions. Chihara uses the relation of satisfaction, while



type theory uses the relation of predication. Otherwise the structural developments of the

theories are basically the same. Hellman’s modal-structural interpretation of set theory

is pattern-equivalent to ordinary set theory, for Hellman posits the possible existence of

a model of the set-theoretic axioms. Thus Hellman’s positions are just whatever objects

exist in the possible-model, and his relations are those that hold in the possible-model;

the set-theoretic structure remains much the same (this being Hellman’s goal, after all).

Field, it is remembered, interprets mathematical assertions literally but regards them

as false. However, his account of how mathematical knowledge is acquired appeals to

modal constructions that very closely resemble Hellman’s modal structural interpretations.

This element of Field’s position involves asserting the possibility of the conjunction of

the axioms of a mathematical theory while at the same time sanctioning certain modal

inferences on the bases of such possibility claims. These modal assertions invoke much of

the structure of ordinary mathematical theories.

Resnik does not claim that structurally equivalent theories share an ontology, at least at

the objects-level. If the ontological component of Shapiro’s objection to modal nominalism

is that modal nominalists are unable to avoid commitment to individual mathematical

objects, then availing himself of Resnik’s account of structure would not be to his benefit.

However, Shapiro never asserts that modal nominalists are committed to the existence of

individual mathematical objects; he does nonetheless claim that modal nominalists are

committed to the existence of structures.36 Resnik (qua realist about structures) agrees with

Shapiro that the modal nominalist is committed to structures, although for slightly different

reasons. Shapiro’s guiding principle is that definitionally equivalent theories share an

ontology. This is bolstered by the accusation that the respective epistemological problems

facing modal nominalism and platonism are commensurate. Resnik is sympathetic only

36At least, he never directly asserts that nominalists are committed to individual mathematical objects.He does suggest this indirectly, however. According to Shapiro’s conception of structure (discussed below),mathematical structures are free-standing objects and are mathematical correlates of ante rem universals. Thepositions of these structures are full-fledged objects. Thus, any theory that is committed to mathematicalstructures is subsequently committed to the existence of the objects which instantiate them.



with the epistemological tactic. In his review of Hellman’s Mathematics Without Numbers,

he writes that,

. . . in most non-mathematical cases it is easier to show that something is possible(or consistent) than to show that it is actual (or true). . . Let us take Hellman’ssimplest case, the possibility of an infinite progression. Given our experiencewith set theory and mathematical logic during this century, we can be confidentthat we cannot deduce this possibility from our knowledge of the finite. . . oncewe set aside our mathematical knowledge, I do not see how we could countthis belief as justified. (Resnik 1992, 118)

What is this mathematical knowledge that he speaks of?

. . . how do we know that infinite progressions are logically possible? Becauseno contradiction follows from the the supposition that they exist, I presume.And how do we know this? Well, we can rest with logical intuitions and ourdeductive experience or we can turn to mathematical models. Historically,we have taken the latter course and have appealed to mathematical objects toclarify intuitive notions of possibility. (ibid., 118)

. . . once we have gone so far as to postulate the possibility of such extraordinaryobjects, I would think that it would be simpler to take them as actual and enjoythe benefits of standard applied mathematics. (ibid., 119)

Here Resnik proffers remarks very similar to those voiced by Shapiro concerning the role

model theory has played in the advancement of knowledge about logical modality. As I

will argue after having described Shapiro’s structuralism, I do not see how this kind of

reply can possibly represent a convincing case for the existence of either structures or more

ordinary kinds of mathematical objects. No one is denying that model theory has allowed

for the fruitful development of modal logic. But model theory alone is incapable of making

any categorical pronouncements about what is possible. The model-theoretic approach

to modal logic is an incredibly useful and rigorous tool for analyzing modal inferences,

but those inferences have to start from somewhere. In model theory, the starting point

is usually set theory; sets are simply assumed to exist. To instantiate Resnik’s account of

structure into Shapiro’s criticism of nominalism would be asking the reader to believe that

modal nominalism is defeated because knowledge of the possibility (and hence actuality)


Shapiro’s Challenge 114 Shapiro and Structure

of a structure is only justified assuming a starting point in model theory. But that would

just be to assume that sets exist in the first place, clearly begging the question against the

modal nominalist. Why should this be lauded an epistemologically superior starting point

when compared with the initial assumption that it is possible for sets to exist?

Ultimately, Resnik’s justification for the existence of sets, models, and patterns is

pragmatic. That, for patterns, possible existence implies actual existence, is admissible only

after it has been conceded that there exist certain initial patterns by way of model theory

and set theory. But the modal nominalist will object at precisely this point, and claim

that the mere possibility of these kinds of objects is sufficient for capturing mathematics.

So Resnik’s account of structure and his criticisms of modal nominalism do not, in the

end, diminish Shapiro’s burden.37 Next I examine Shapiro’s own structuralist theory. The

upshot is rather anticlimactic, given what has just been said in response to Resnik; Shapiro

winds up deferring to set theory in the same way and for the same reasons as Resnik, and

consequently he makes no compelling case against the modal nominalist.

2.6.2 Shapiro and Structure

Shapiro does not share Resnik’s resistance to providing a formal theory of structure. Resnik

resists doing this because he does not believe that mathematical theories literally quantify

over structures. On this point, Shapiro and Resnik are in clear disagreement:

My outlook towards this [structure] language (and theory) may be calledworking realism. Classical logic, impredicative definition, the axiom of choice,and extensionality are freely employed. There is no apology for this language,and no major regimentation is envisioned. That is, structures are not thoughtof as reinterpreted versions of anything else. As I see it, structures are partof the ship of Neurath, and the structuralist language is useful for describingother parts of the vessle [sic]—including the semantics of mathematical theories.(Shapiro 1989b, 162)

Moreover, Shapiro believes that there are strong ties between the language, logic, and

ontology of mathematical theories:37Resnik even admits that it is possible to be, “a structuralist without being a realist about mathematical

objects” (1997, 270).



. . . my view is that the sorts of ‘objects’ studied by a branch of mathematics—thesorts of structures—are determined by the allowed constructions and sanc-tioned inferences. In short, the logic and the objects share a common source—the moves available to the ideal constructor. (1989a, 32)

Shapiro forges these connections in conjunction with his belief that second-order logic

underlies the practice of mathematics.38 Although in first-order logic there are no categori-

cal theories with infinite models (due to the Lowenheim-Skolem Theorem), nevertheless

there are such theories in second-order logic. The categoricity of second-order theories

comports well with the structuralist manifesto that mathematicians are only concerned

with structure up to isomorphism. For better or worse I am not much interested in ex-

amining whether Shapiro is correct in holding that second-order logic is the underlying

logic of mathematics.39 As in Resnik’s case, I am mostly interested in determining whether

Shapiro’s conception of structure is at all capable of validating his criticisms of modal

nominalism.

Shapiro begins by introducing the idea of a system. A system is a collection of objects

with certain relations (Shapiro 1997, 73). Shapiro’s favorite example of a system is a baseball

defense: a collection of nine baseball players in some particular spatial arrangement at a

ballpark. Other examples include extended families, arrangements of chess pieces, and

symphonies. Structures are defined in relation to systems: A structure is the abstract form of

a system (ibid., 74). This definition alone is not entirely helpful. For what, if anything, is the

abstract form of a system? According to Shapiro, these are to be understood as objects akin

to ante rem universals; he goes as far as to label his view “ante rem structuralism.” Thus,

for Shapiro, ante rem structures are freestanding objects—objects that exist independently

of their exemplifications. Moreover, he also views the places or positions of structures as

bona fide objects (ibid., 83).

For two systems to exemplify the same structure it is sufficient, but not necessary, that

the systems be isomorphic (ibid., 91). Shapiro also admits Resnik’s criterion of structure-38A view he defends in detail in (Shapiro 1991).39For discussion, see (Shapiro 2005) and (Jane 2005).



equivalence as capturing a more coarse-grained notion of when two systems instantiate

“essentially the same” structure. And of course, Shapiro’s own notion of definitional

equivalence provides an account of sameness of structure for systems. However, Shapiro

diverges sharply from Resnik in constructing an axiomatic theory of structure. To do this

Shapiro develops structuralist correlates of the set-theoretic axioms:

Infinity: There is at least one structure that has an infinite number of places.

Subtraction: If S is a structure and R is a relation of S, then there is a structureS ′ isomorphic to the system that consists of the places, functions, andrelations of S except R. If S is a structure and f is a function of S, thenthere is a structure S ′′ isomorphic to the system consisting of the places,functions, and relations of S except f .

Subclass: If S is a structure and c is a subclass of the places of S, then there is astructure isomorphic to the system that consists of c but with no relationsand functions.

Addition: If S is a structure and R is any relation on the places of S, thenthere is a structure S ′ isomorphic to the system that consists of the places,functions, and relations of S together with R. If S is a structure and f isany function from the places of S to places of S, then there is a structureS ′′ isomorphic to the system that consists of the places, functions, andrelations of S together with f .

Powerstructure: Let S be a structure and s its collections of places. Then thereis a structure T and a binary relation R such that for each subset s′ ⊆ sthere is a place x of T such that ∀z(z ∈ s′ ≡ Rxz).

Replacement: Let S be a structure and f a function such that for each placeof x of S, fx is a place of a structure, which we may call Sx. Then thereis a structure T that is (at least) the size of the union of the places in thestructures Sx. That is, there is a function g such that for every place z ineach Sx there is a place y in T such that gy = z.

Coherence: If Φ is a coherent formula in a second-order language, then thereis a structure that satisfies Φ.

Reflection: [For any first- or second-order Φ in the language of structure the-ory:] If Φ, then there is a structure S that satisfies the (other) axioms ofstructure theory and Φ. (ibid., 93-5)

The most curious of these axioms is almost certainly the Coherence axiom (also known as

the “coherence principle”). According to the Coherence axiom, if a formula in a second-

order language is coherent, then there exists a structure that satisfies it. But why should



anyone suppose that the Coherence axiom is true—that the coherence of a formula suffices

for the existence of a structure that satisfies it? And what does it mean in the first place to

say that a formula is coherent?

Doubts about the truth of the Coherence axiom are likely to stem from adherence to the

almost universally held belief that possibility does not imply actuality, even for structures.

One possible baseball defense involves each outfielder crowding the left-field foul line and

each baseman laying prostrate at a distance behind their respective bases determined in

inches by the closest number of the Fibonacci sequence to the number on their jersey, with

the shortstop running figure-eights in uneven paces in shallow center. A colleague that

will go unnamed assures me that this is a legal defense in Major League Baseball, but to

both his knowledge and mine, no manager has ever adopted it. So the abstract form of this

possible system has no actual representatives. If possibility implies actuality for structures,

and the system in question is possible, then the system must have an actually existing

abstract form.40 And for Shapiro, the positions of structures are bona fide objects. So if the

structure in question exists, then so do its positions. But by Shapiro’s own admission, a

baseball defense is a type of structure that can be realized only by having baseball players in

its positions. Without players there can be no defense. Is there no avoiding the awkward

conclusion that the mere possibility of such a baseball defense necessitates the existence of

a group of players that have at least once implemented it?

A very similar situation arose above in conjunction with Shapiro’s conception of defini-

tional equivalence outlined earlier in the chapter. Shapiro claims that any two definitionally

equivalent theories have identical ontological commitments. In response, Chihara gestures

at the possibility of constructing two theories—one about cats, the other about dogs—that

meet Shapiro’s conditions for definitional equivalence. The awkward conclusion in that

case was that certain cats are identical to certain dogs. But that criticism was not terri-

bly bothersome, for Shapiro does not claim that definitionally equivalent theories have

40Thanks to Larry Powers for pointing out an error in an earlier articulation of this point.



identical ontological commitments in general but only in the case of mathematical theories.

Moreover, Shapiro is concerned mostly with the ontological commitments a theory has due

to the structure that it invokes, as opposed to the ontological commitments that a theory

has in virtue of the ranges of its first-order quantifiers. My ultimate suggestion was that

Chihara’s response was unsuccessful barring a more detailed investigation of Shapiro’s

views on ontology. I remind the reader of this earlier component of the dialectic because

I believe a similar strategy might allow Shapiro to avoid the awkward conclusion of the

previous paragraph. Pace Resnik, Shapiro can grant that no one should believe that, in

general, the possibility of a structure is evidence of its actual existence. Nevertheless there

may be reason to believe that as far as mathematical structures are concerned, there is no

appreciable gap between the possible and the actual. But this maneuver requires for its

success two further steps. First, it must be shown how it is possible to determine, about a

particular structure, whether it is of the mathematical or non-mathematical variety. Sec-

ond, reason must be given for supposing that even in the case of mathematical structures,

possible existence suffices for actual existence.

Shapiro believes that he has hit upon a relevant distinction between mathematical and

non-mathematical structures:

In mathematical structures, on the other hand, the relations are all formal, orstructural. The only requirements on the successor relation, for example, arethat it be a one-to-one function, that the item in the zero place not be in its range,and that the induction principle hold. No spatiotemporal, mental, personal, orspiritual properties of any exemplification of the successor function are relevantto its being the successor function. (ibid., 98)

A system does not count as a baseball defense unless its meets the non-formal requirement

that its positions be filled by entities capable of playing baseball. Meanwhile, provided

that there are a denumerable infinity of them, any old objects can count as a system the

abstract form of which is the natural number structure. The relations of a baseball defense

are restricted in that they can only be filled by a certain class of spatiotemporal objects. No

such restriction at all is made by the successor relation. This qualifies the natural number



structure as formal. Formality, then, is the criterion by which a structure’s actual existence

can be inferred from its possible existence. But through which means can it be determined

whether a relation is formal?

If each relation of a structure can be completely defined using only logicalterminology and the other objects and relations of the system, then they are allformal in the requisite sense. (ibid.)

And what does Shapiro mean by ‘logical terminology’?

. . . the present proposal is that a relation is formal if it can be completely definedin a higher-order language, using only terminology that denotes Tarski-logicalnotions and the other objects and relations of the system, with the other ob-jects and relations completely defined at the same time. All relations in amathematical structure are formal in this sense. (ibid., 99)

One hesitates to rest so much on a choice of the logical vocabulary. I happen to have the

intuition that logic is lordly and so is not in the business of making existential claims. But

if Shapiro is to be trusted, mathematical ontology is relative to the selection of logical

vocabulary, because logic decides what is formal, and what is formal decides what exists

in mathematics. For better or worse, Shapiro does not appear share my views on the

lordliness of logic. This is a very high-level disagreement—Shapiro’s structuralism is

wedded to a picture of mathematics according to which language, logic, and ontology are

deeply intertwined—a picture that I am not at all interested in defending. So if pressed,

here is where I would dig my trenches. Unfortunately, there is not room for this battle

in this dissertation, so I must pass over matters in relative silence. Thus, for the sake of

argument, I am willing to grant Shapiro his connections between language, logic, and

ontology. As I shall argue in the sections to come, even with this concession Shapiro is not

able to make a convincing case against modal nominalism.

That mathematical structures are characterized exclusively by formal relations helps to

distinguish them from non-mathematical structures. So Shapiro’s quest is not yet exposed

as unsatisfiable. Nevertheless he still faces the task of explaining why, even in the case



of mathematical structures, possible existence suffices for actual existence. That is, he

must still justify the truth of the Coherence axiom. And, of course, he still faces the task of

explaining what it means in the first place to say that a structure is possible or coherent.

Evidence for the coherence of a theory of a structure can come in a variety of forms:

Simple abstraction and pattern recognition, linguistic abstraction, and implicit definition.

Shapiro claims that small finite structures can be apprehended directly via pattern recogni-

tion or simple abstraction. For instance, the various manifestations of the letter ‘E’ (e, e,

E , E ) can all be recognized as such despite the fact that, “there is nothing like a common

shape to focus on” (ibid., 114). What these various ‘E’s do have in common, according

to Shapiro, is position in an alphabet structure. Thus learning the function of the letter

‘E’ involves learning about a particular kind of structure. However, pattern recognition

and abstraction are only capable of illuminating small, finite structures. For large finite

structures, and for structures up to the size of the continuum, one must enlist the aid of

linguistic abstraction. With linguistic abstraction, one can describe large finite patterns. For

example, linguistic tokens can evince structural relations of very large numbers that could

not be recognized via pattern recognition. Using numerals it is a trivial task to distinguish

the 999,9999 cardinal structure from the 1,000,000 cardinal structure. However, few, if

any humans, could quickly distinguish a circular array of 999,9999 dots from a circular

array of 1,000,000 dots. To extend matters to the infinite, it can be observed that there is no

natural limit to the size of a cardinal (or ordinal) structure. Linguistic abstraction permits

the coherent description of denumerably infinite structures, and structures constructible

from denumerably infinite structures (e.g., the rationals, Cauchy sequences of rationals,

etc.). A key point underlying this discussion is the idea that simple abstraction, pattern

recognition, and linguistic abstraction all provide the resources for giving coherent descrip-

tions of structures. That structures exist is a further claim that is inferred only after the

Coherence axiom is assumed (ibid., 118; 120).



Finally, a set of axioms can be thought of as implicitly defining a structure. The axioms

can then be subjected to deductive study. One is justified in believing that a theory of a

structure is coherent if and only if, after a rigorous examination, no contradictions are

derived from its axioms. Implicit definition is really the most important notion here:

Many interesting mathematical structures, especially in set theory, involve large infinite

cardinalities, which implies that pattern recognition and linguistic abstraction will never be

capable of justifying beliefs in the coherence of many mathematically interesting structures.

But what does the coherence of an implicit definition come to? Shapiro admits that

coherence is not to be understood as deductive consistency; ante rem structuralism is a

creature of second-order logic, and there is no completeness proof in second-order logic

(PA + ¬Con(PA) is deductively consistent, but lacking models, it is arguably not coherent).

Shapiro would prefer coherence to be thought of an analogue of satisfiability, but the

notion of satisfiability involves a circularity; satisfiability is a model-theoretic notion:

Normally, to say that a sentence Φ is satisfiable is to say that there exists amodel of Φ. The locution “exists” here is understood as “is a member of theset-theoretic hierarchy,” which is just another structure. What makes us thinkthat set theory itself is coherent/satisfiable? (ibid., 135)

Shapiro’s ultimate move is to concede that coherence is an undefinable primitive:

. . . there is no getting around this situation. We cannot ground mathematics inany domain or theory that is more secure than mathematics itself. All attemptsto do so have failed, and once again, foundationalism is dead. The circle thatwe are stuck with, involving second-order logic and implicit definition, is notvicious and we can live with it. I take “coherence” to be a primitive, intuitivenotion, not reduced to something formal, and so I do not venture a rigorousdefinition. (ibid.)

Shapiro, like Resnik, requires noninferential belief in the coherence of some mathematical

structures. For Shapiro, that structure is set theory:

In mathematics as practiced, set theory (or something equivalent) is taken tobe the ultimate court of appeal for existence questions. Doubts over whether acertain type of mathematical object exists are resolved by showing that objectsof this type can be found or modeled in the set-theoretic hierarchy. (ibid., 136)



Once set theory is in place—that is, once sets are assumed to exist—coherence can qualify

as a criterion for the existence of structures that are constructible out of sets. But whither

the coherence of set theory?

Surely, however, we cannot justify the coherence of set theory itself by modelingin the set-theoretic hierarchy. Rather, the coherence of set theory is presupposedby much of the foundational activity in contemporary mathematics. Rightlyor wrongly (rightly), the thesis that satisfiability is sufficient for existenceunderlies the background for model theory and mathematical logic generally.Structuralists accept this presupposition and make use of it like everyoneelse, and we are in no better (and no worse) of a position to justify it. Thepresupposition is not vicious, even if it lacks external justification. (ibid.)

The coherence of set theory, then, is a basic presupposition of mathematics. Even a modal

nominalist can agree with that. But coherence qualifies as a criterion of existence only

after it is assumed that sets exists. No reason is given to suppose that, for set theory itself,

coherence implies existence. If Shapiro thinks otherwise he has conflated two importantly

distinct things: The status of the Coherence axiom on the supposition that sets exist; and

whether the coherence of set theory implies the existence of sets. As I argue below, it is

not clear that Shapiro is entitled to use the Coherence axiom to establish the existence of

ante rem structures, even presuming that sets do indeed exist. Moreover to make the leap

and claim that sets do indeed exist is to go well beyond what is literally “presupposed” in

mathematics—which makes Shapiro’s platonism about structures rather difficult to square

with things he says earlier:

. . . at no time did the mathematical community don philosophical hats and de-cide that mathematical objects—numbers, for example—really do exist. . . (ibid.,25)

One cannot “read off” the correct way to do mathematics from the true ontology,nor can one “read off” the true ontology from mathematics as practiced. Thesame goes for semantics, epistemology, and even methodology. (ibid., 34)

What, then, is his source of warrant for supposing that sets do indeed exist? What is the

justification for his platonism?



I present an account of the existence of structures, according to which an abilityto coherently discuss a structure is evidence that the structure exists. . . theargument for realism is an inference to the best explanation. (ibid., 118)

The reader is asked to believe that sets and structures exist because doing so best explains

various aspects of mathematics and its practice. What makes the actual existence of

sets and structures the best explanation here? That story is captured by his criticism

of modal nominalism: Shapiro sees the epistemological difficulties stemming from the

postulation of abstract mathematical objects as the principle obstacle to his structuralist

platonism. He thinks that these difficulties translate into epistemological difficulties for

modal nominalists stemming from their appeal to assertions about what is primitively

possible. And, of course, he thinks that the structural similarities between set theory and

the various modal nominalist accounts of mathematics exemplify the same structures.

And since structure is all that matters as far as mathematical ontology is concerned, modal

nominalists are just as ontologically burdened as structuralists. In the end, he claims that

there is simply not much to choose from:

The fact that any of a number of background theories will do is a reason toadopt the program of ante rem structuralism. Ante rem structuralism is moreperspicuous in that the background is, in a sense, minimal. On this option, weneed not assume any more about the background ontology of mathematicsthan is required by structuralism itself. (ibid., 96)

Left unexamined throughout this discussion thus far is what justification Shapiro has

for holding that, as far as mathematics goes, structure is all that matters. Shapiro’s primary

contention against the modal nominalist comes with his proposal that

. . . definitional equivalence serve[s] as a criterion of the formal strength of modaland nonmodal theories and. . . that this notion be used as an indication that theintended structures, and thus the ontology/ideology of different theories, arethe same. If T is definitionally equivalent to T ′, then neither is to be preferredto the other on ontological/ideological grounds. (ibid., 242)

On the same page he refers to (Wilson 1981) as furnishing a justification for the idea that

definitional equivalence has these curious powers. It should be worthwhile to assess



Wilson’s evidence on matters.

Wilson claims to be able to show that,

. . . the ontology of a mathematician is to be determined by the formal propertiesof the structure he postulates, not by the names he employs in the descriptionof it. The claim is that any two theories meeting the conditions of severalparagraphs back41 share an ontology, I shall call structuralism, since this thesisrepresents a plausible reading of the jingle “Mathematics is only interested instructure up to isomorphism” (ibid., 414).

As an example, he considers two versions of set theory: One of these is ZF, containing only

pure sets. The other is ZP, which is identical to ZF except that it contains natural numbers

as urelements. Wilson imagines presenting both theories to a group of mathematicians

and asking them if they believe that ZF is ontologically impoverished when compared to

ZP. He hypothesizes that,

Many (or most) mathematicians would probably demur, arguing that the on-tology of ZF actually does include the numbers, etc., because it includes anω-sequence and methods for building the needed sets from it. (ibid., 413-4)

This is offered as convincing evidence that,

. . . if one accepts a theory of a certain formal strength, one cannot deny it itsstandard ontology, no matter in what syntactic guise its assertions may appear.(ibid., 419)

Wilson draws the conclusion that ZF and ZP have identical ontological commitments

because he thinks that mathematicians would argue that both include the numbers. But

what designation is to be attached to the phrase ‘the numbers’? If “the numbers” are

supposed to be the traditional platonist’s mathematical objects, then according to Wilson,

the structuralist position on arithmetic comes to the surprisingly non-structuralist slogan

that: “Any ω-sequence contains the numbers as objects.” But this appears to me to be a

41“. . . we shall only be concerned with interdefinable theories T , T ′ which claim to represent the totalontology for mathematics and whose intended structures S (T ) and S (T ′) are such that if a structure S isbuilt from S (T ) based upon the definitions T provides for the terms of T ′, S will be isomorphic to S (T ′)”(ibid., 413).


Shapiro’s Challenge 125 Reply to the Structuralist Response

misleading interpretation of what mathematicians would mean in saying that ZF and ZP

both contain the numbers. An interpretation that is more in the spirit of structuralism holds

that the differences between ZF and ZP are rather uninteresting, not for any ontological

reasons, but rather because as far as mathematics goes, either version of set theory is

adequate for capturing number theory. But the attendant slogan here is not that “any

ω-sequence contains the numbers,” but instead the ontologically innocuous claim that

“any ω-sequence satisfies the properties that are important to number theorists.”

Wilson’s evidence is altogether weak; that formally similar theories have identical

ontological commitments depends upon a controversial understanding of what math-

ematicians might say in the context of a thought experiment. I conclude that Shapiro’s

deference to Wilson is misplaced. Wilson’s remarks on structure and ontology lend very

little credence to Shapiro’s claim that definitionally equivalent theories share an ontology.

Frankly, it is not clear to me what justification there is for propounding structuralism

as a view that says that, as far as mathematical objects go, structure is all that matters.

I do not object to structuralism as a view that highlights the kinds of relationships that

mathematicians find important and interesting; in that regard I think the view is superior

to many other philosophical accounts of mathematics. However, Shapiro is happy to admit

that mathematicians have never decided that mathematical objects exist. In light of that I

do not see how he finds it admissible to claim that as far as mathematical objects go, there is

much of anything that matters! As a view about the ontology of mathematics, structuralism

receives very little—if any—support from mathematical practice.

2.7 Reply to the Structuralist Response

The Structuralist Response holds that modal nominalists are committed to the existence of

mathematical objects—viz., structures—because modal nominalist accounts of mathematics

implicitly characterize coherent mathematical structures. The following argument captures

the basics of this response:


Shapiro’s Challenge 126 Reply to the Structuralist Response

10. Modal nominalists assume that theories of mathematics are primitively possible (or

coherent).

11. Shapiro’s Coherence axiom; any coherent formula in a mathematical language is

satisfied by some actually existing structure.

12. By (10) and (11), formulae of modal nominalist theories of mathematics are satisfied

by actually existing structures.

13. By (12), commitment to modal nominalist theories of mathematics necessitates com-

mitment to actually existing structures.

14. By (13), modal nominalist theories do not succeed in eschewing commitment to

mathematical objects.

The Structuralist Response also has the capacity to bolster Shapiro’s claim that modal

nominalism and platonism are on an equal epistemological footing:

15. The modal nominalists defended here maintain that an epistemologically defensible

account of mathematics must assume no more than the primitive possibility, or

coherence, of mathematical theories.

16. By (11) and (15), any evidence the modal nominalist provides for the coherence

of mathematical theories can be appropriated by the platonist as evidence for the

existence of mathematical structures.

17. Since actuality implies possibility, any evidence the platonist has for positing the

existence of mathematical structures can be appropriated by the modal nominalist as

evidence for the coherence of the appropriate mathematical theories.

18. By (16) and (17), modal nominalism and platonism are on an equal epistemic footing.


Shapiro’s Challenge 127 Withering Coherence

Finally, the Structuralist Response can make sense of Shapiro’s accusation that modal

nominalism is inferior a propos of justifying modal assertions because modal nominalists

accept primitive modal notions.

19. The modal nominalists defended here rely on primitive modality.

20. Platonists can reduce modal concepts set-theoretically.

21. By (19) and (20), platonists are better equipped than modal nominalists for justifying

modal assertions.

If sound, these arguments are incredibly damaging to the modal nominalist approach. It

is fairly clear that in the first two arguments, premise (11)—Shapiro’s Coherence axiom—

does all of the important work—and the bulk of the remainder of this chapter is dedicated

to demonstrating that (11) should be rejected. I shall eventually argue that the third

argument is invalid—that platonism can provide a reductive account of modality does not

show that platonism provides a means for justifying modal assertions. But this response

involves prospecting more broadly issues related to reductionism about modality, and so I

shall delay developing it in much detail until the next chapter (although I shall outline my

thinking on matters at the end of this chapter).

2.7.1 Withering Coherence

Note that Shapiro accepts coherence as a primitive notion. Modal nominalists, meanwhile,

require primitive modality. Thus the battle over primitives is between coherence and

primitive modality. But these two notions are essentially no different from one another.

Would it make sense to call a theory coherent, but not possible (at least using whichever

sense of the word ‘possible’ that nominalists have in mind)? Or vice versa? Here, anyway,

I am thinking of the theories that characterize the structures of accepted mathematics,

of which modal nominalists and platonist both seek accounts. If that is right, then as

far as primitives go, there is not a whole lot of difference between Shapiro’s account of

mathematics and the various modal nominalist theories I defend in this dissertation.



The main difference between Shapiro and modal nominalists is Shapiro’s adherence to

his Coherence axiom; that any coherent second-order formula is satisfied by some actually

existing structure. Hellman approaches this axiom with a degree of skepticism:

Why should coherence suffice for existence of mathematical structures whereasin virtually any other domain of inquiry, coherence does not suffice for exis-tence? (2001b, 196)

Shapiro’s unsatisfying response is that the Coherence axiom holds for structures the

relations of which are exclusively formal. According to Harold Hodes, this maneuver raises

some important unanswered questions:

Is formality an absolute property of relations (as the surrounding materialsuggests), or is it relative to systems? And if the latter, to which systems? Oris there only one relevant system? and if so what is it? Perhaps that structureitself? What resources are allowed for this definition in a higher-order language?Without answers to these questions, Shapiro’s explanation of formality remainsunilluminating. (2002, 470)

But even if Shapiro can be alloted an intuitively palatable conception of formality, the

general worry remains. Criticism comes from a number of sources. Hellman elsewhere

writes,

[The Coherence axiom] can be thought of as a post-Godelian substitute forformal consistency: the axiom mimics Hilbert’s idea that consistency sufficesfor mathematical existence. But of course it is not a formal notion and seemsno clearer than a primitive notion of (second-order) logical possibility—indeed,perhaps less so, for do we have anything as developed as modal logic govern-ing “coherent?” And if we identify these notions, then the Coherence Axiomappears even more problematic, for why should mere logical possibility sufficefor existence? Indeed, why not just rest with the former. . . ? (2005, 546)

Izabela Bondecka-Krzykowska and Roman Murawski ask,

Can one claim that a structure defined by an implicit definition, hence by a setof axioms, does exist by appealing to the consistency of the axioms and to thecompleteness theorem. . . ? No “normal” mathematician is doing this. . . If suchmethods were rejected so where from should we know then that structuresdefined by implicit definitions do exist? (2006, 36-7)



And Fraser MacBride suggests that,

. . . even if coherent categorical descriptions are guaranteed to be non-empty,there still remains an epistemological issue about how it can be establishedthat descriptions are coherent and categorical. For if we cannot know whichdescriptions of structures are coherent and categorical, then there is little epis-temic comfort to be had from the reflection that such descriptions—whichever,without our knowing, they may be–are inevitably satisfied. (2008, 162)

As seen in the last section, Shapiro has a response to these concerns. He claims that

existence questions in mathematics can be deferred to set theory. If set theory is itself

coherent and if sets do indeed exist, much of the mystery surrounding the Coherence

axiom is removed. MacBride finds a certain irony in Shapiro’s deference to set theory:

It is also noteworthy that the ontology of ante rem structuralism performsno substantial role in this, the most fundamental component of Shapiro’sepistemology for mathematics. For when there is a need to establish whether adescription is coherent and categorical, structures drop out of sight, and it isquestions of set existence that come to the fore. (ibid., 163)

Ironical or not, is Shapiro’s flight to set theory an acceptable move? In a recent paper,

Shapiro says that he is, “content to have my account, as a whole, judged alongside other

philosophies of mathematics on the overall score of what does best in accounting for

mathematics, and its role in our intellectual and personal lives, using whatever resources

are available for this endeavor” (2011, 138). This comes as a result of his acknowledgment

that he

. . . cannot deduce the coherence principle from non-mathematical premiseswhich any opponent will accept. Nor do I accept the burden of justifying thecoherence principle on such grounds. The coherence principle is part of anoverall philosophy of mathematics which, I claim, well explains the enterprise.(ibid., 147)

This holistic inference to the best explanation is worth unpacking.

Shapiro’s evidence for the Coherence axiom is alleged to come from the foundational

activity within mathematics itself. It is worth repeating a passage already quoted from

before:



In mathematics as practiced, set theory (or something equivalent) is taken tobe the ultimate court of appeal for existence questions. Doubts over whether acertain type of mathematical object exists are resolved by showing that objectsof this type can be found or modeled in the set-theoretic hierarchy. Exam-ples include the “construction” of erstwhile problematic entities, like complexnumbers. (Shapiro 1997, 136)

As written, this passage does contain an argument for platonism:

22. Set theory serves as a court of appeal for existence questions.

23. Existence questions about Xs are resolved by showing how to construct Xs in the

set-theoretic hierarchy.

24. Therefore, if Xs can be constructed in the set-theoretic hierarchy, then Xs exist.

Three points are in order. First, the inference from (22) and (23) to (24) is ampliative. The

suggestion appears to be that because set theory has served as the court of appeal for

existence questions in recent professional mathematical practice, it will continue to do

so for other existence questions. Second, this argument only establishes platonism if it is

granted in advance that sets exist. If sets do not exist, then the fact that Xs can be modeled

in the set-theoretic hierarchy is not evidence that Xs exist. Third, and finally, this argument

is not by itself sufficient to establish the existence of ante rem structures, even if the first

two concerns can be addressed. When constructing models in set theory, one is always

constructing particular mathematical objects, or particular isomorphism types—one is not

directly constructing ante rem structures. If sound, this argument establishes a form of

objects-platonism, and not necessarily a form of a platonism about structures, potentially

undermining the ability of the Coherence axiom to expose the would-be structural commit-

ments of modal nominalist theories. Nevertheless, what this suggests is that the principle

embodied by (24) is not clearly equivalent to Shapiro’s Coherence axiom (line (11) from

the arguments above). Shapiro even appears to recognize that (24) is not equivalent to the

Coherence axiom:



To “model” a structure is to find a system that exemplifies it. If a structure isexemplified, then surely the axiomatization is coherent and the structure exists.Set theory is the appropriate court of appeal because it is comprehensive. Theset-theoretic hierarchy is so big that just about any structures can be modeledor exemplified there. (ibid.)

Here, ‘model’ is in quotes, suggesting that the “modeling” of a structure is not the same

kind of activity as the modeling of complex numbers. This is made clear by his indication

that what things are actually constructed in set theory are systems or exemplifications of

structures, and not the structures themselves. Thus the inference that there exist ante

rem structures is something that is to occur over and above whatever constructions are

completed within the set-theoretic hierarchy.

These three points raise some important questions: First—what is the actual role of

a principle like (24) in the mathematical community, and does its role lend support to

using the Coherence axiom in a critique of modal nominalism, or in an argument for the

existence of ante rem structures? Second—what is the evidence for the existence of sets?

And third—supposing that (24) and the Coherence axiom are indeed distinct, what other

sort of justification can be provided for Shapiro’s Coherence axiom? I will examine each of

these questions in turn.

2.7.1.1 Satisfiability and Mathematics

What is the mathematical role of (24)? And does its role support using the Coherence

axiom to criticize modal nominalism and to argue for the existence of ante rem structures?

Shapiro is on record as claiming that something like (24) is at work in mathematics.

Indeed, he claims that it “underlies mathematical practice” (ibid.). Under what conditions

is it appropriate to appeal to the Coherence axiom for producing answers to existence

questions? For Shapiro, whenever a theory is sufficiently formal, the existence of a structure

can be inferred from the ability to coherently discuss the theory. Thus the scope of the

Coherence axiom is not limited just to the existence questions that have actually arisen in

mathematics, or even just to those existence questions that would likely arise in practice;



it applies generally to sufficiently formal theories. He acknowledges this as a point of

contact with Mark Balaguer’s full-blooded platonism:

Mathematical objects are tied to structures, and a structure exists if there is acoherent axiomatization of it. A seemingly helpful consequence of this is that ifit is possible for a structure to exist, then it does. Once we are satisfied that animplicit definition is coherent, there is no further question concerning whetherit characterizes a structure. Thus, structure theory is allied with what Balaguercalls “full-blooded platonism” if we read his “consistency” as “coherence.”(ibid., 134)

However, mathematicians are typically not interested in just any consistent extension of set

theory, nor are they interested in just any consistent theory of a sufficiently formal stripe. If

any sort of coherence-implies-existence principle is used in mathematics, it is restricted just

to theories that serve a recognized goal in mathematics. David Corfield makes a related

observation a propos of set theory:

While it is impressive enough to be able to represent more or less any desiredconstruction, it has a problem in that it does not know how to say ‘No’. Itcannot distinguish between those constructions that the mathematically literatewill realise are patently pointless and those that stand at least some chance ofgainful employment. (2003, 239)42

I take it that the best possible evidence for using the Coherence axiom to establish the

existence of ante rem structures (and to criticize modal nominalism) would involve showing

that the existence of ante rem structures matters to mathematicians in some important way,

or that mathematicians themselves use the Coherence axiom to establish the existence

of ante rem structures. But mathematicians appear unconcerned with establishing the

existence of ante rem structures; mathematicians have not packed ante rem structures in their

luggage for their voyage on Neurath’s ship. I take this as evidence that Shapiro’s use of

the Coherence axiom is not straightforwardly analogous to the relatively uncontroversial

uses of (24) (for establishing, e.g., that complex numbers exist). That is, the practice of

mathematics lends no direct support to the use of the Coherence axiom for establishing the

42See also (Maddy 1997, 202).



existence of ante rem structures. What sort of indirect support is available to Shapiro will

be discussed shortly.

2.7.1.2 The Existence of Sets

What is the evidence for the existence of sets? The argument from (22) and (23) to (24) is

most plausible as an argument for a reductionist variety of objects-platonism, according to

which mathematical objects can be reduced to sets. But, as I have argued previously, it is

only convincing evidence for platonism if it is granted in advance that sets exist. Although

Shapiro does not argue directly for the existence of the set-theoretic hierarchy, he does

come close to doing so in the following passage:

I take it that I can assume the correctness of standard accepted mathematicsalong the way, and I can assume that mathematicians know the bulk of standard,accepted mathematics. . . My use of set theory at this juncture is based on thebelief that set theory itself is coherent, a belief shared by mathematicians. . . Itfollows from the coherence of set theory that if we show that any proposedimplicit definitionD is satisfiable, thenD is itself coherent, and thusD describesa structure, or a class of structures. (Shapiro 2011, 149)

Although perhaps not intended as such, this nevertheless is not convincing evidence that

sets exist. Shapiro claims that the coherence of set theory is a presupposition of mathematics.

But, by itself, coherence is just a primitive notion akin to second-order satisfiability. No claim

follows about the existence of sets unless it is assumed in addition, and as a presupposition of

mathematics, that the Coherence axiom applies to set theory. But Shapiro does not claim the

application of the Coherence axiom to set theory as a presupposition of mathematics (and

nor should he, given that mathematicians do not use the principle in the first place)—he

claims only that mathematics presupposes the coherence of set theory. But if all that is

known is that set theory is coherent, then all that strictly follows is that if sets exist, then

anything coherently describable in set theory would exist as well.

If Shapiro’s criticism of modal nominalism (along with his case for the existence of

ante rem structures) depends on this detour through set theory, then ante rem structuralism

is at risk of begging the question against nominalists, who are willing to grant at most



that set theory is coherent. In the present setting, the criticism that modal nominalists are

committed to structures only has force if it is granted that either (a) sets exist, or (b) that

the Coherence axiom applies to set theory. Option (a) is clearly question-begging. Option

(b) is equivalent to option (a), and moreover, is circular in the context of exploring whether

anyone is justified in appealing to the Coherence axiom. The question naturally arises as

to whether this circularity is vicious. Shapiro claims that it is not:

. . . ante rem structuralism is itself no more secure than is set theory. . . So if I werelooking to provide some sort of extra-mathematical justification or security forset theory, making use of set theory would be viciously circular, and structureswould have dropped out of the picture. The grounding would take place in settheory. But that is not my game. (ibid.)

The security of set theory, platonism about sets, and the existence of ante rem structures

all seems to hinge on just what is presupposed in mathematical practice. I reiterate that

the mere coherence of set theory is not sufficient grounds for extracting the ontological

results that must be extracted in order to establish the existence both of sets and of ante

rem structures. Moreover the mere coherence of set theory is not sufficient to use these

(unavailable) results to figure in Shapiro’s criticism of modal nominalism, since that

criticism requires the Coherence axiom. This would settle matters, save for the fact that

Shapiro also musters indirect, holistic support for the Coherence axiom and more generally

for his ante rem structuralism, and it is to these considerations that I now turn.

2.7.1.3 Holistic Justifications for the Coherence Axiom

So, given that (24) and the Coherence axiom are distinct principles, what sort of justification

is left for the Coherence axiom? Reflecting on the burdens he faces in defending ante rem

structuralism, Shapiro writes that he is, “content to have my account, as a whole, judged

alongside other philosophies of mathematics on the overall score of what does best in

accounting for mathematics, and its role in our intellectual and personal lives, using

whatever resources are available for this endeavor” (ibid., 138), and that, “[t]he coherence

principle is part of an overall philosophy of mathematics which, I claim, well explains the



enterprise” (ibid., 147). He further remarks that he is out to, “provide a justification for a

philosophical interpretation of mathematics,” and that such activity, “is not a deductive

enterprise, where I would have to start with non-mathematical, self-evident principles”

(ibid., 149). These are points often reiterated in the the fourth chapter of (Shapiro 1997)—

that the application of the Coherence axiom to ante rem structures is part of a highly

plausible account of the overall mathematical enterprise. But what are his reasons for

supposing that his ante rem structuralism is more plausible than its competitors?

Unfortunately, what lies behind these plausibility considerations are nothing other

than the criticisms of nominalism described above in §2. One of these criticism is that

modal nominalists do not eschew commitment to mathematical objects because they

share in the structural commitments of mathematical theories. But why should a modal

nominalist (or anyone else) feel compelled to recognize and acquiesce to the structural

commitments of a mathematical theory? One possible reply says that if modal nominalist

theories are to be successful they must, at a minimum, provide coherent descriptions of

mathematical theories, and thus via the Coherence axiom, they must acknowledge that

there exist structures described by these theories. But it has already been established that

this use of the Coherence axiom goes beyond the presumably uncontroversial uses of a

principle like (24). Whether anyone is justified in so using the Coherence axiom is precisely

what is at issue. But without assuming that the Coherence axiom can be used to establish

the existence of ante rem structures, I have difficulty understanding why someone such as a

modal nominalist should be willing to grant that her theories are ontologically committed

to ante rem structures.

I think this poses a dilemma regarding the potential justifications for using the Coher-

ence axiom in a criticism of modal nominalism. On the one hand, it can be maintained

that it is just a fact about definitional equivalence that definitionally equivalent theories

have identical ontological commitments, at least when it comes to mathematical theories.

On the other hand, it can be maintained that two mathematical theories have identical



ontological commitments when both (a) the theories are definitionally equivalent to one

another, and (b) both theories are sufficiently formal so that the Coherence axiom applies

to them. The first option is undesirable because in order for it to be the case that def-

initionally equivalent theories share an ontology, it must be supposed that one means

something entirely different than most logicians and mathematicians when one speaks of

‘definitional equivalence,’ and this undermines the idea that, so justified, one’s conception

of definitional equivalence is consistent with or supported by the practice of mathematics

(where such qualities are presumed desiderata). The second option is circular—but is it

viciously circular? Shapiro’s desired result is that the Coherence axiom can be used to

establish the existence of ante rem structures, and so bloat the modal nominalist’s universe.

This principle is embedded in an overall account of mathematics which is alleged to be

more plausible than the alternatives, in large part because the alternatives are ontologically

on a par. But that the alternatives are ontologically on a par involves inferring, e.g., that

because modal nominalist theories posit coherent descriptions of mathematical theories,

modal nominalist theories are committed to the existence of ante rem structures. This kind

of use of the Coherence axiom is no different in kind from the uses of it that stand in

need of justification, so this response, if advocated, would indeed be viciously circular.

And it is unclear whether there are any other reasons why it should be supposed that,

for instance, modal nominalists must recognize the alleged structural commitments of

their views. Absent such reasons, Shapiro is not entitled to claim that alternative views

are ontologically on a par with ante rem structuralism, vitiating premise (11) in the above

arguments.

But suppose Shapiro is right that the primitive notion of coherence by itself plays an

important foundational role in mathematics (in the sense that the coherence of set theory is

a basic presupposition of mathematics). Since structuralists, “accept this presupposition

and make use of it like everyone else, and [are] in no better (and no worse) of a position to

justify it” (ibid., 136), it would appear to follow that all interested philosophical parties are


Shapiro’s Challenge 137 What Exactly is the Problem?

privy to the presuppositions of mathematical practice. But I cannot help but recognize how

little conceptual distance there is between a primitive notion of coherence and, for example,

a primitive notion of logical possibility. It would indeed make little sense to maintain

that the coherence of set theory is a presupposition of mathematics, but then argue that

some additional evidence is required for licensing belief in the primitive logical possibility of

set theory. Indeed, there seems to be little to no additional conceptual space between the

notions of primitive logical possibility and satisfiability than there is between coherence

and satisfiability—here it is worth recalling that Shapiro takes structural similarities to

be indicators of ideological similarities. If that is right, then mathematics itself provides

all of the evidence certain modal nominalists require in order to be justified in asserting

the primitive logical possibility of mathematical theories. This suggests to me that the

scaffolding of ante rem structures is an eliminable component of philosophical interpre-

tation of mathematics, since philosophical accounts of mathematics can get well enough

along with just a notion like coherence (or primitive logical possibility), without needing

something like the Coherence axiom. This means that a view incorporating the Coherence

axiom faces a burden not shared by the alternatives—that of justifying the Coherence

axiom itself. Although these considerations by no means settle the dispute between modal

nominalism and platonism, I do take them to show that there is room for preferring certain

modal nominalist views on ontological grounds.43 In any case, the burden of justifying the

Coherence axiom is a burden that remains for ante rem structuralism irrespective of one’s

attitude toward modal nominalism. Let me expand on these remarks in the next section.

2.8 Shapiro’s Challenge: What Exactly is the Problem?

Shapiro levies several accusations against modal nominalist accounts of mathematics.

The first of these is that modal nominalist accounts of mathematics raise epistemological

questions that are just as serious as those facing platonism. A second is that the modal

nominalist languages ultimately do not avoid reference to mathematical objects. The first

43Thanks to Eric Hiddleston for suggesting an important clarification here.



accusation arises because modal nominalist accounts of mathematics invoke primitive

modality and require undefended assumptions about what is possible or what it is possible

to do. The second accusation follows from Shapiro’s structuralist views; modal nomi-

nalist theories are definitionally equivalent to set theory, and subsequently share in set

theory’s ontological commitments. The upshot is a not-so-balanced tradeoff: Platonists are

burdened by a vast ontology, whereas modal nominalists are burdened by both primitive

modality and by a vast ontology.

I have argued that the mere fact that two theories are definitionally equivalent lends no

support to the claim that such theories must have identical ontological commitments. I

then considered two alternative ways of reaching this conclusion. One alternative holds

that modal nominalist theories are no more than synonymous paraphrases of mathematical

languages. If two sentences are synonymous, then they both mean the same thing, and

so cannot differ in their ontological commitments. This alternative makes the mistaken

assumption that modal nominalist theories are to be regarded as mere paraphrases of ordi-

nary mathematical languages. A second alternative delves further into the structuralist’s

conception of mathematical objects. According to Shapiro’s conception of structure, one

is justified in believing that a structure exists provided that one has good evidence that

the structure is coherent. Modal nominalist theories posit what is essentially the logical

possibility (or coherence) of mathematical theories, which via Shapiro’s Coherence axiom,

commits these theories to the existence of mathematical structures. This second alternative

ultimately begs the question against modal nominalism, for in order for the coherence of

a mathematical structure to imply its existence, it must be assumed that the Coherence

axiom applies to set theory, which is equivalent to the assumption that sets exist.

At best, the tradeoff between modal nominalism and platonism is the following: Pla-

tonists are burdened by a vast ontology, whereas modal nominalists are burdened by

primitive modality. And, if the reader is keeping track of primitives, the tradeoff is actually

balanced in favor of nominalism: Platonists (a la Shapiro) accept both the modal notion of



coherence and the Coherence axiom as basic postulates, whereas modal nominalists only

require primitive modal notions. And it is doubtful that there are any important differences

(epistemological, ontological) between the logical modalities modal nominalists use and

the notion of coherence Shapiro invokes. The upshot is that modal nominalists require

only primitive modality, whereas Shapiro requires primitive modality and the contentious

assumption that for some kinds of objects—mathematical structures—possibility implies

actuality. I should think this welcome news for modal nominalists.

A third accusation lurks in the background, and that is that modal nominalists are

somehow particularly burdened by their invocation of primitive modality. Why? Because

once the platonist has available the assumption that sets exist, she can avail herself of the

set-theoretic reduction of the modal notions. According to the platonist, one can explain

what is logically possible by appealing to one’s knowledge about sets. So the problems

she faces in coming to terms with how she can have knowledge about the coherence of

mathematical structures are just the same problems she faced at the outset—of how she is to

come to know that sets exist and to know various things about them. The modal nominalist,

meanwhile, has no obvious route for explaining or justifying the modal assertions that

figure in her theories. Shapiro’s Challenge to the nominalist is this: For the modal nominalist

to show that she can, on nominalistically acceptable grounds, explain how it is possible for her to

justify the modal assertions that figure in her theories, and to explain, again on nominalistically

acceptable grounds, why she is entitled to apply the results of modal logic when constructing and

applying her theories. As I have explained previously, my focus is on the former component

of the challenge. As I argue in the next chapter, this component presents itself to the modal

nominalist as more than a merely epistemological challenge. Rather, it presents two related

tasks: To provide an account of the content of modal claims, and to provide an account

of how to justify claims about this content. The ironic result, according to Shapiro, is that

platonists are actually in a better position to complete both tasks.

It is unclear whether the platonist’s purported ability to generate a reductive account



of modality is really the boon that Shapiro makes it out to be. For what is important

in the dispute at hand is not the status of the modal notions in general, but rather the

status of particular claims of possibility (or coherence). For Shapiro to present an effective

objection to modal nominalism, he must assume that some reductive account of modality

genuinely illuminates the epistemology of modality (this assumption is contested in the

next chapter). Suppose it is granted that Lewisian possible worlds provide a reductive

base for the metaphysical modalities. Then it is incontestable that p is possible just in case

p is true in some possible world. Thus, the question about whether a statement p really

is possible can be answered by determining whether there is some possible world in which

p is true. Unfortunately this does nothing to reduce the mystery about how knowledge

that ♦p is acquired, indeed, it appears to add to it; now one must claim to know about the

particular constitution of some possible world. An analogous situation arises in the case of

the set-theoretic reduction of the logical modalities. First, it must be granted that sets exist

and that it is possible for humans to know various things about them (this, of course, is

the selfsame epistemological problem platonism faces at the outset). But now knowledge

that ♦p is knowledge that there exists a model that satisfies p, i.e., it is knowledge about

some particular construction in set theory. If logical possibility is read as consistency,

then this knowledge can be acquired by constructing a consistency proof of p. However,

not even Shapiro can benefit from such particular claims of possibility. Shapiro requires

particular claims of coherence. And if the logic in question is second-order (as it must be

for Shapiro), consistency is no guarantee of coherence. But serving as the foundation for

consistency proofs is the only means by which the set-theoretic reduction can assist humans

in determining which formulae (and theories) are coherent and which formulae (and

theories) are not coherent. The problem is that without a reductive theory of coherence,

Shapiro is in no better position than anyone else for justifying particular claims of coherence.

I suppose one could posit “coherence proofs” as a novel proof-theoretic concept, akin to

consistency proofs, but which guarantee the coherence of their conclusions. But that would



presuppose some method of distinguishing coherence proofs from consistency proofs.

Since Shapiro does not say what coherence is, this strategy is not likely to benefit him in

any way.44

Shapiro is also on record as claiming that pre-theoretic intuitions about modality are

much too vague to support the particular claims of possibility that modal nominalists

make (1993, 475). Against this, Chihara writes (on behalf of Constructibility Theory),

Let us consider the development of finite cardinality theory given [in Con-structibility Theory]. Notice that the exposition and discussions of this theory(including all the proofs of theorems) are given without any appeals to anymodel-theoretic notions or to results from possible worlds semantics. Deduc-tions and inferences are made using modal reasoning and without any mentionof set theory or set-theoretical results. Yet, it can be seen that standard theoremsof number theory can be obtained within this system. (2004, 205)

Chihara’s point is that he never explicitly utilizes any set-theoretic results in developing

cardinality theory. It is true that his development mirrors previous platonist developments,

and it is plausible that had there never been any previous developments of cardinality

theory that Chihara would never have happened upon his own development of cardinality

theory. But to say that this means that Constructibility Theory is deeply enmeshed in set

theory—including the purported ontological commitments of set theory—is to commit the

genetic fallacy. As Chihara is quick to remind the reader, reasoning in mathematics did not

begin with set theory (ibid.). In principle, the historical development of mathematics could

have occurred using a modal framework. Modal locutions would then be the dominant

vehicles for making and evaluating mathematical assertions. In this hypothetical situation,44An alternative possibility involves establishing the coherence of mathematical theories using infinitary

logics. Infinitary logics allow for infinite conjunctions and disjunctions, as well as (in many cases) quan-tification over infinitely many variables. Let λ, κ be cardinals, with λ ≤ κ. An infinitary base languageL(κ, λ) permits conjunctions and disjunctions of sets of formulae of cardinality ` where ` < κ, and permitsquantification over sets of variables of cardinality where < λ. The “largest” infinitary language thatpossesses the completeness property is L(ω1, ω). See (Keisler and Knight 2004, 18) and (Bell 2012). L(ω1, ω)can be used, with the completeness property, to demonstrate the coherence of PA. However, Shapiro remarksthat L(ω1, ω) cannot even characterize the reals up to isomorphism (1991, 241). Thus, even in infinitarylogics, the coherence of most mathematical structures cannot be demonstrated via consistency proofs, thanksto the failure of completeness for languages “larger” than L(ω1, ω). In any case, Shapiro does not appear tobe terribly sympathetic to infinitary logics (ibid., 240), and so there is reason to doubt that he would acceptthem as a means for demonstrating the coherence of mathematical theories.



it might be complained that the objects of platonist set theory—sets—are only clearly

understood because of the prior work done in modal mathematics, and that set theorists

would not have been able to construct anything resembling a useful mathematical theory

without depending on the prior insights of modal mathematics. Shapiro’s anti-nominalist

argument fairs no better than this anti-platonist argument. I take it that both are equally

bad, and ought to be rejected.45

These concerns aside, I understand Shapiro to have made an important criticism of

modal nominalism. Modal nominalists must say something about why they are justified in

making modal assertions, and they must say something about why this places them in a

less onerous position when compared to platonism. To give some account of the flavor

of this challenge, consider what Bob Hale has said about the epistemological burdens of

Hellman’s Modal Structuralism:

It is easy—but an error—to slide. . . into supposing that acceptability is, quitegenerally, the default status for possibility claims—that such claims are in-variably to be presumed innocent until proven guilty, and stand in no needof defence unless and until a case is made against them. . . it is far from evi-dently true that there could be a complete ω-sequence of concrete objects: it isaccordingly reasonable to require grounds for belief that it is so. (Hale 1996,132)

Confidence that arithmetic is consistent, however well founded, goes no way atall towards resolving the present issue—for all we so far know or have reasonto believe, it may be that, as a matter of necessity, arithmetic admits of no concretemodels. (ibid., 134)

Hale can be understood as demanding of the modal nominalist that she elicit some kind of

evidence for her possibility claims—evidence that must avoid implying or assuming the

existence of mathematical objects.

How is a modal nominalist to respond to Shapiro’s and Hale’s demands? Mary Leng

lights the one way while discussing the epistemology of set theory:

45Cf. my discussion of this issue in chapter three, §5.



. . . if we look for direct reasons for believing in the consistency of set theory,or in the consistency of particular models that set theory provides for othermathematical theories, it is plausible that this problem is somewhat moretractable than the problem of finding reasons for believing that set theory istrue. (2007, 105)

According to Leng, such “direct reasons” can arise in two important ways. In the first

place, inductive considerations can be mustered in support of the consistency of set theory:

So far we’ve not come across a derivation of a contradiction from the axioms of(for example), ZFC, despite many mathematicians working on set theory (and,indeed, looking for constructions that might produce a contradiction) for manyyears. (ibid.)

An alternative form of inductive justification comes from applications:

. . . the best explanation of the applicability of a mathematical theory requiresonly that that theory is consistent with the non-mathematical facts (and there-fore that it is consistent per se). But if the best explanation of the successfulapplication of a piece of mathematics requires the mathematical theory that weapply to be consistent, then an application of inference to the best explanationwould provide an inductive justification for our belief in the consistency of thattheory. (ibid., 106)

Second, arguments from intuition can support possibility claims:

. . . arguments from intuition are poor arguments for the truth of mathematicaltheories. However, our ability to conceptualize configurations of objects thatsatisfy the axioms of set theory or number theory might well provide us withsome evidence that those theories are consistent. (ibid.)

Leng reminds the reader that the iterative conception of sets, “appears to us to involve

no contradiction,” and claims on behalf of arithmetic that, “our ability to conceive of an

ω-sequence seems to provide particularly good reason for believing the consistency of PA”

(ibid.). The upshot is that,

At the very least, such considerations suggest that the prospects for defendingconsistency claims on such grounds are greater than the prospects of defendingclaims concerning the face-value-truth of the theories we are considering. (ibid.,107)



Presumably all of this is the selfsame evidence that someone such as Shapiro would

offer on behalf of the claim that set theory and arithmetic are true theories. Does that

unveil platonism and modal nominalism as residing in a comparable justificatory position?

I should think not. Again, Shapiro forges the the link between the consistency of a

mathematical theory and the truth of a mathematical theory via his Coherence axiom. But

his Coherence axiom is only in a position to secure this result if (a) set theory is coherent,

and (b) the Coherence axiom applies to set theory. Together (a) and (b) are equivalent to the

assumption that set theory is true. But whether set theory is true (or whether it is merely

consistent) is what is at issue. Moreover, defending (b) presents a burden for the platonist

that is not shared with the modal nominalist.

Nevertheless Shapiro can still claim that platonism fairs better at explaining how one

justifies assertions about the coherence or consistency of mathematical theories. And his

support for such a claim comes to the pronouncement that modal nominalists must accept

primitive modality, whereas platonists can avail themselves of the set-theoretic reduction of

the logical modalities. This strategy raises important questions about what a reduction of

modality consists in and whether reductive theories of modality are capable of performing

the justificatory work that Shapiro claims that the set-theoretic reduction is capable of

performing. In the next chapter I argue that the set-theoretic reduction is not capable of

assisting the platonist in justifying assertions about what is logically possible. (And nor is

a reduction like Lewis’s capable of assisting someone in justifying assertions about what is

metaphysically possible.) If modal-mathematical assertions are unjustified under modal

primitivism, then they become no less unjustified under a platonistic reduction of modality.

It follows from this that modal nominalists are not particularly burdened by their use

of primitive modality. But then modal nominalists and platonists like Shapiro are in a

similar position with respect to justifying assertions about the coherence or consistency

of mathematical theories. And so, on Shapiro’s picture, modal nominalists are the least

burdened, because they are not forced to defend dubitable views about the relationship



between the consistency of mathematical theories and the existence of mathematical

objects.


146

Chapter 3

Reducing Modality as a Solution to Shapiro’s Challenge

No one shall expel us from the Paradise that Lewis has created.

—Anonymous

I wouldn’t dream of trying to drive anyone out of this paradise. . . I would try to show you that it is

not a paradise—so that you’ll leave of your own accord.

—Ludwig Wittgenstein

3.1 Introduction

Shapiro’s objection to modal nominalism purports to show that modal nominalists both

(a) do not succeed in evading the ontological commitments of ante rem structuralism, and

(b) encounter serious difficulties on account of their use of modality. I have shown in the

previous chapter that Shapiro is not successful in demonstrating that modal nominalist

accounts of mathematics share in the ontological commitments of ante rem structuralism,

vitiating (a). The general goal of this chapter is to explore the significance the remaining

component (b) of Shapiro’s objection to modal nominalism. Precisely what is problematic

about the modal nominalist’s use of primitive modality? Can these problems—whatever

they may be—be avoided by appealing to a reductive account of modality? In particular,

does Shapiro successfully avoid these problems through his appeal to the set- or model-

theoretic reduction of the logical modalities? The main contention of this chapter is that

Shapiro is unsuccessful in showing that the modal nominalist, by appealing to primitive

modality, faces particularly challenging obstacles concerning the justification of modal

assertions.

In section two I provide a recapitulation of Shapiro’s Challenge with the aim of precisely

eliciting what is troubling about the modal nominalist’s use of primitive modality. The


Reducing Modality 147 Introduction

issue for the modal nominalist (qua modal primitivist) is that she is incapable of giving

content to the modal assertions that figure in her theories of mathematics, in the sense that

she does not provide an account of the truth-conditions for these assertions. Without such

an account, the modal nominalist appears unable to provide any compelling justifications

for the modal assertions that figure in her theories. Thus, modal assertions under modal

nominalism threaten to be completely unjustified, and perhaps, unjustifiable. Meanwhile,

the platonist can appeal to the set-theoretic reduction of the logical modalities in order to

provide truth-conditions for modal assertions, including those made by modal nominalists.

Thus the platonist can at least provide an account of what makes these assertions true or

false, making it possible to identify what challenges her modal epistemology faces (e.g.,

knowledge about what kinds of sets exist and what can be modeled in set theory).

In section three I set precedent for my response to Shapiro through a discussion of

David Lewis’s well-studied reduction of modality. Lewis argues that assertions about

what is metaphysically necessary and possible can be reduced to assertions about what is

true in a vast realm of spatiotemporally disconnected universes—Lewis’s possible worlds.

Lewis proposes that knowledge about what transpires in merely possible worlds can be

justified on the basis of already justified commonsense modal intuitions, together with the

further metaphysical hypotheses Lewis makes about the nature of his worlds. What is of

significance here is that his analyses of metaphysical possibility and necessity in terms

of spatiotemporally disconnected universes (i.e., possible worlds) serve only as a rewrite

rules for translating back and forth between ordinary modal assertions and assertions

about what transpires in possible worlds. The analyses themselves confer no justification

upon lone modal assertions or upon lone assertions about what transpires in possible

worlds. Thus, the mere fact that Lewis is in possession of a reductive account of modality

provides no reason for supposing that Lewis, when compared to a primitivist about the

metaphysical modalities, is in a preferable position with respect to his ability to justify

modal assertions. This conclusion is not in conflict with anything Lewis has said about his


Reducing Modality 148 Introduction

reduction of modality, but it does raise the prospect of applying similar reasoning against

Shapiro. Perhaps the analyses of the logical modalities in terms of models and sets does

not in fact place set-theoretic reductionists, when compared to modal primitivists, in a

preferable position a propos of justifying assertions about what is logically possible.

As I explain in section four, the conclusion of section three applies to possible worlds

style reductions of modality more generally. Possible worlds style reductions—that is,

reductions that take advantage of basic Kripkean ideas about modal semantics—share

a common structure: A non-empty universe or domain of ϕs is postulated; necessity is

analyzed as truth in all ϕs, possibility as truth in a ϕ, etc. What differentiates one such

reduction from another is the particular metaphysical resources it substitutes for ϕ. The

analyses themselves are again merely rewrite rules and confer no justification upon lone

modal assertions or upon lone assertions about what is the case in or according to the

non-empty domain of ϕs.

In section five I argue that the general conclusion of section four applies in the particular

case of the set-theoretic reduction of the logical modalities. The basic idea of the set-

theoretic reduction is that assertions about what is logically possible can be reduced to

claims about what can be modeled in set theory (though as I explain in the section, this

reduction may only be useful for claims about the logical consistency of mathematical

theories and about the possible existence of mathematical objects). I explain that the

analyses of the logical modalities in terms of sets, like Lewis’s analyses of the metaphysical

modalities in terms of worlds, serve only as rewrite rules for translating back and forth

between assertions about what is logically necessary or possible and set existence assertions.

The analyses themselves perform no work justifying lone modal assertions or lone set

existence assertions. Thus, it would be wrong to insist that Shapiro is in a privileged

position a propos of justifying logical possibility assertions, solely in virtue of the fact that

he can appeal to a reductive theory of the logical modalities.

Actually, it is possible to show that Shapiro’s situation is at least as undesirable as the


Reducing Modality 149 Reduction and Shapiro’s Challenge

situation facing the modal nominalist (qua modal primitivist). Set existence claims, for

Shapiro, ultimately reduce to claims about which sets can be coherently described, and

Shapiro’s notion of coherence is a primitive notion that is indistinct from a primitive notion

of logical possibility. If modal nominalists are unjustified in using modal assertions, then it

stands to reason that Shapiro is no less unjustified in using coherence assertions. On the

other hand, if Shapiro is justified in using coherence assertions (and he believes that he is in

virtue of his contention that the coherence of set theory is a presupposition of mathematics),

then, given the indistinctness of coherence and logical possibility, there seem to be no

obstacles preventing the modal nominalist from using modal assertions.

In section six I consider some reasons for doubting that modal primitivism is nominalis-

tically acceptable. A viable modal primitivism appears committed to the idea that there is

something to which modal operators or modal properties apply, and it seems unlikely that

the primitivist can guarantee that this something is nominalistically acceptable. However,

I contend that the modal nominalist is still entitled to the response to Shapiro that I offer in

section five—that Shapiro has given no reason for supposing that the problems nominalists

face by using primitive modality can be avoided by appealing to the set-theoretic reduction.

But what might these problems be? To answer this question it should be helpful to revisit

Shapiro’s Challenge.

3.2 Reduction and Shapiro’s Challenge

Recall from the previous chapter that I left the epistemic component of Shapiro’s objection

to modal nominalism largely intact. What is this criticism? Loosely, that whenever the

platonist faces questions such as “how do we know that p?” (where p involves reference to

mathematical objects), the modal nominalist faces questions such as “how do we know that

♦p?” According to Shapiro, “it is hard to see how adding primitive possibility operators to

the formation of epistemic problems can make them any more tractable,” which implies

that, “it is hard to see how the [nominalist] has made any progress over the [platonist] on



the sticky epistemic problems” (1997, 226). Thus, the modal nominalist appears saddled

with seemingly intractable epistemological questions about what is categorically possible

(or coherent, or constructible, etc.). Related to this concern is Shapiro’s hypothesis that the

modal nominalist’s use of primitive modality is (perhaps unwittingly) facilitated by her

understanding of set theory and model theory. It is not clear why the modal nominalist is

entitled to use familiar systems of modal reasoning in application to her particular modal

primitives. Shapiro claims that modal nominalism engenders an “epistemic loss” because,

“it is hard to see what grounds our antirealists would use to support the modal assertions,

given that they do not believe in models” (ibid., 237). Consider what Shapiro says against

Hellman’s modal structuralism:

. . . because Hellman is out to drop the realist perspective, it is not clear why he isentitled to the traditional, model-theoretic explications of the modal operatorsof logical necessity and logical possibility. For example, the usual way ofestablishing that some sentence is possible is to show that it has a model. ForHellman, presumably, a sentence is possible if it might have a model (or if,possibly, it has a model). It is not clear what this move brings us. (ibid., 229)

Shapiro, by contrast, holds that set theory is, “the source of the precision we bring to

modal locutions” (ibid., 232). Thus, the platonist does have grounds for supporting the

modal assertions that she makes, viz., the set- or model-theoretic reduction of the logical

modalities. A provisional account of Shapiro’s Challenge, then, is that modal nominalists

must somehow demonstrate that they have nominalistically acceptable grounds for the

modal reasoning they employ and the modal assertions which figure in their theories. (As

I have indicated previously in the dissertation, I take Chihara and Field to have offered a

response to the charge that modal nominalists lack nominalistically acceptable grounds for

employing modal reasoning—my concern here is with the charge that modal nominalists

lack nominalistically acceptable grounds for the modal assertions which figure in their

theories.)

What is not clear is whether this criticism of modal nominalism is accurately described

as a purely epistemic criticism. Consider Shapiro’s use of the terms ‘grounds’ and ‘sup-



porting’ in the following sentence, quoted above: “it is hard to see what grounds our

antirealists would use to support the modal assertions, given that they do not believe in

models” (ibid., 237). It should be noted that Shapiro is not using the term ‘grounds’ in a

technical or formal way (sensu (Rosen 2010) or (Fine 2012)), but rather as a synonym for

‘resource.’ So the complaint is that it is unclear what resources modal nominalists might

use for supporting the modal assertions they make. But this complaint is still ambiguous

between a metaphysical and an epistemic reading.1 On the epistemic reading, to have

grounds or resources for supporting an assertion means something like having a suffi-

cient reason for believing that the assertion is true. On the metaphysical reading, to have

grounds or resources for supporting an assertion means something like being able to iden-

tify and describe the content of the assertion using more basic principles. This ambiguity

runs throughout Shapiro’s “epistemic” criticism of modal nominalism. Nevertheless the

epistemic and metaphysical readings are related in a prima facie way—presumably having

the metaphysical resources for supporting an assertion p—that is, understanding what p

means, what it is that makes claims like p true, etc.—goes a long way towards generating

the epistemic resources for supporting p—that is, having a sufficient reason for believing

that p (which could come, for instance, from prior confidence that the truth-conditions for

p obtain).

Below I will often be concerned with the justification of modal assertions. Let me

stipulate now that my use of ‘justify’ throughout the remainder of this chapter primarily

concerns the epistemic sense of “grounds” just described. That is, I shall principally be

concerned with the question of whether, e.g., modal nominalists have sufficient reasons

for believing that the modal assertions they use are true—any unmodified instance of

‘justify’ (or any instances modified by epistemic vocabulary) should be assumed to have

this epistemic connotation. When my concern is specifically with the attribution of content

or truth-conditions to modal assertions, I will either say so directly, or use modified

1Thanks to Eric Hiddleston for pointing out this ambiguity.



expressions such as ‘metaphysical grounds,’ or ‘metaphysically justify.’ Thus, to say that

someone is metaphysically justified in asserting p (or to say that someone has metaphysical

grounds for p) is to say that this person is capable of limning the truth-conditions for p.

What is the contribution of primitive modality in support of Shapiro’s contention that

the modal nominalist has no resources for supporting the modal assertions which figure in

modal nominalist theories? Recall from the previous chapter that Shapiro takes the alleged

fact of the definitional equivalence between set theory and modal nominalist theories to

be itself sufficient evidence for placing an epistemic burden on the modal nominalist a

propos of her use of modality. This gives the appearance that a fundamental issue for the

modal nominalist is that she is epistemically unjustified in making modal assertions. Indeed,

Shapiro presents this charge as standing independently from the claim that set theory is the

“source of the precision” of the modal notions. However, and perhaps in distinction from

Shapiro’s actual dialectic, I would like to suggest that the modal nominalist’s purported

lack of epistemic justification is entirely derivative of the modal nominalist’s (qua modal

primitivist) lack of metaphysical justification for modal assertions.2

Notice that the claim that set theory is the “source of the precision” of the modal

notions is naturally interpreted as saying that the platonist has resources for justifying

modal assertions according to the metaphysical reading—rather than as a claim that the

platonist has epistemic resources for justifying modal assertions. What the set- or model-

theoretic reduction provides is an account of the truth-conditions of logical possibility

claims.3 These metaphysical resources provide epistemic gains in a certain sense—they

help the platonist to determine just what modal knowledge consists in—e.g., knowledge

2On this point let me be clear: I am not disputing that Shapiro objects to modal nominalism on accountof its apparent inability to ground modality in both epistemic and metaphysical senses (though he wouldlikely describe both senses as ‘epistemic’)—it is only that it is not obvious to what degree Shapiro takes theseobjections to be related. It seems consistent with his criticism that these objections are less-closely relatedthan I maintain below.

3I will describe the set-theoretic reduction in greater detail in §5, but the following characterization shouldbe sufficient for the moment: The set-theoretic reduction essentially holds that the set-theoretic hierarchyexists and what can be constructed in the set-theoretic hierarchy produces all possible models. The notionsof logical possibility and consistency can then be analyzed as truth in a model; the notion of logical necessitycan be analyzed as truth in all models.



about sets and what is or can be modeled in set theory. The platonist’s metaphysical

resources, then, help her to gain traction on just how intractable (or not) her modal

epistemology is. Meanwhile the modal nominalist, qua modal primitivist, provides no

metaphysical resources for justifying the modal assertions she makes. That is, the modal

nominalist deprives herself of the ability to provide an account of the truth-conditions of

modal assertions. This situation does represent an “epistemic loss” in the sense that now

the modal nominalist seems incapable of even circumscribing the basic form of her modal

epistemology, and consequently, she is not able to make reliable judgments about what

epistemological difficulties she might face (if any) and how these difficulties compare to

those faced by platonistic reductionists. This problem seems more accurately described

as a metaphysical problem as opposed to an epistemic one.4 The worry is that modal

nominalists (qua modal primitivists) are unable to provide an account of what it is that

makes modal assertions true or false. This does, however, suggest a related epistemic

concern—if modal nominalists cannot explain what it is that makes modal assertions true

or false, how could they ever have sufficient reasons for believing that modal assertions

are true?

But given that modal epistemology under the set-theoretic reduction simply reduces to

the epistemology of platonist set theory, it seems premature to claim, as Shapiro suggests,

that the set-theoretic reductionist is in a preferable position when compared to the modal

primitivist. That is because it is doubtful that the set-theoretic reductionist can show

that assertions about the existence and character of the set-theoretic hierarchy are any

better epistemically justified under platonism than are modal assertions under modal

nominalism.5 What, then, should be made of Shapiro’s “epistemic” objection to modal

nominalism? I argue below that little is ultimately to be made of this criticism because the

problems modal nominalists encounter when advancing claims about what is primitively

possible are not clearly worse than the problems platonists encounter when advancing

4Thanks to Eric Hiddleston for this observation.5Of this, more later.


Reducing Modality 154 Mission Planning

claims about what can be modeled in the set-theoretic hierarchy. The ultimate goal of

this chapter, then, is to undermine the idea that the set-theoretic reduction of the logical

modalities, solely in virtue of its reductive character (i.e., solely in virtue of its capacity

to metaphysically justify modal assertions), provides a means for epistemically justifying

assertions about what is logically possible. If that is right, i.e., if the set-theoretic reduction

itself does no epistemic-justificatory work, then it would be misleading to criticism modal

nominalism on the grounds that modal nominalists, qua modal primitivists, employ

epistemically unjustified (or unjustifiable) modal assertions.

3.2.1 Mission Planning

Unfortunately it is not clear how best to proceed with this discussion. One reason for

this is that Shapiro does not explain in much detail how the set-theoretic reduction is

supposed to help illuminate the epistemology of the logical modalities, but instead appears

content to observe that, “[w]e inherit the language/framework [of modal terminology in

its application to mathematics], with the connection to set theory already forged” (1997,

238). What this suggests is that assertions about what is logically possible are likely for

Shapiro to be justified, both metaphysically and epistemically, along the same lines as

he proposes to justify beliefs about the existence of ante rem structures: That structures

exist allegedly follows from the coherence (Shapiro’s own modal primitive) of descriptions

of structures. The existence of the set-theoretic hierarchy is ultimately demonstrated by

positing that the set-theoretic hierarchy can be coherently described, perhaps by the ZFC

axioms. But this strategy would lead to a clear circularity if used to justify the set-theoretic

reduction: Why is λ logically possible? Because λ can be modeled in the set-theoretic

hierarchy. Why does the set-theoretic hierarchy exist? Because set theory is coherent.

Recalling here that coherence is a modal primitive should be sufficient for explaining why

the previous explanation is circular, but perhaps somewhat more can be said. Shapiro

describes coherence as an informal correlate of the notion of satisfiability, and further, that

coherence can be explicated model-theoretically (ibid., 135). But there is virtually nothing


Reducing Modality 155 Mission Planning

that distinguishes coherence from a primitive notion of (second-order) logical possibility

(which can similarly be “explicated” model-theoretically). For example, the content of the

assertion that ‘ZFC is coherent’ is not appreciably distinct from the content of the assertion

that ‘ZFC is (second-order) logically possible.’ (That these notions can be explicated model-

theoretically is of little help because for Shapiro the existence of models derives from the

coherence of set theory.) So if Shapiro is right that modal nominalists are metaphysically

and epistemically in the dark about logical possibility, then there would appear to be

serious room for doubting that Shapiro himself is any less in the dark (metaphysically and

epistemically) when it come to coherence. And further, given the purported reductive

relationships between modality and set theory, and between set theory and coherence,

Shapiro would also be in the dark about logical possibility. (More on this in §5.)

A second reason why it is not clear how to proceed from here is that questions are

seldom raised about the use of set theory as an appropriate vehicle for justifying assertions

about what is logically possible. However, there is another reduction for which questions

of justification have often been discussed—David Lewis’s reduction of the metaphysical

modalities to possible worlds. There are many structural affinities between the set-theoretic

reduction of the logical modalities and Lewis’s possible worlds reduction of the meta-

physical modalities. For instance, each reduction is usually proposed as vindicating the

core semantical and inferential properties attributed to necessity and possibility under S5

systems of modal logic (in which the accessibility relation between “worlds” is reflexive,

transitive, and symmetric). In the next section I shall argue that Lewis’s view presents

a clear example of the idea that reductive theories of modality, solely in virtue of their

reductive characters (i.e., solely in virtue of their ability to metaphysically justify or ground

modal assertions), do not have any advantages over modal primitivism when it comes

to epistemically justifying modal assertions. In §4 I identify the structural elements of this

reasoning, which I then apply in §5 to the set-theoretic reduction of the logical modalities.


Reducing Modality 156 Lewis’s Reduction

3.3 Justifying Modal Assertions Under Lewis’s Reduction

Perhaps no theory of modality has attracted more attention than the modal realism elabo-

rated and defended by David Lewis. According to Lewis, in addition to the actual world,

there exist countless other possible worlds of (nearly) all shapes and sizes—“every way

that a world could possibly be is a way that some world is” (1986, 2). Lewis’s worlds

are offered as fodder for a reduction of the metaphysical modalities—to be metaphysically

possible to is be true in a possible world; to be metaphysically necessary is to be true in

all possible worlds—and such talk of possible worlds is to be construed literally. After

describing the basic working parts of Lewis’s reduction, I show that the reductive elements

of modal realism—the analyses of metaphysical possibility and necessity in terms of possi-

ble worlds—serve only as rewrites rules, and confer no justification upon lone assertions

about what is metaphysically possible or upon lone assertions about what transpires in

possible worlds. Thus, Lewis’s reduction of the metaphysical modalities is not in itself

a useful implement for justifying assertions about what is metaphysically necessary or

possible. For the remainder of this section the reader should assume, unless told otherwise,

that the modalities under discussion are the metaphysical modalities.

3.3.1 Lewis’s Reduction

The salient features of Lewis’s account are as follows: A possible world is the mereological

sum of a series of spatiotemporally related individuals.6 No two spatiotemporally unre-

lated individuals form parts of the same world. A principle of recombination holds that for

any two individuals, there is a possible world containing spatiotemporally related qualita-

tive duplicates of both individuals.7 Actuality is indexical. The following truth-conditions

apply for de dicto modal assertions:

1. Possibly-p is true iff p is true in a possible world.

6For a succinct account of the mereological principles required, see (Maguire, forthcoming).7“Size and shape permitting.” See (Lewis 1986, 89-90).


Reducing Modality 157 Lewis’s Reduction

2. Necessarily-p is true iff p is true in every possible world.

Lewis’s account also purports to provide truth-conditions for de re modal assertions. For

Lewis, individuals exist in one and only one world—transworld individuals (at least,

transworld concreta) are not countenanced. Instead, he invokes intransitive transworld

similarity relations known as counterpart relations. For x and y belonging to distinct

worlds, x is a counterpart of y just in case nothing in x’s world more closely resembles

y than x itself (x may not be unique in this regard, and there may be worlds containing

nothing similar enough to y to count as one of y’s counterparts). The truth-conditions for

de re modal assertions may now be stated:

3. y is possibly-P iff for some counterpart x of y, x has P .

4. y is necessarily- or essentially-P iff for every counterpart x of y, x has P .

In addition to providing a reductive account of the metaphysical modalities, Lewis’s

modal realism is alleged to provide parsimonious accounts of counterfactual conditionals,

supervenience, propositions, and properties. For instance, a proposition can be identified

with the set of all possible worlds in which it is true; similarly, a property can be identified

with the set of all individuals that have the property. In Lewis’s words, his universe of

possible worlds is a “philosopher’s paradise.”

That Lewis’s modal realism possesses all of these allegedly useful qualities is offered as

an inference to the best explanation style argument for the existence of possible worlds,

and, more generally, for the truth of Lewis’s overall theory of modality.8 I must confess to

a large degree of skepticism about this justification for modal realism—both about the idea

8Lewis’s appeal to parsimony is somewhat ironic, to say the least, since Lewis’s ideological parsimonycomes at the cost of an extravagant ontology. Lewis claims that this increase in ontology is quantitative ratherthan qualitative (1973, 87), but this illusion evaporates when one’s attention is drawn to the denizens ofLewis’s worlds. Charles Chihara writes that, “Lewis does think there are unicorns, spirits, and all manner ofstrange creatures. The ontological commitments of his doctrines are as heavy as any theory any philosopher I knowhas ever espoused. This is a very high theoretical price to pay for something as nebulous and questionableas a parsimonious set of primitives” (1998, 102). There seems to be little to recommend the idea thatconsiderations of parsimony support Lewis’s modal realism (unless one is contrasting it with an even moreontologically extravagant view!).


Reducing Modality 158 Lewis and Justifying Modal Assertions

that Lewis’s account genuinely provides the best overall account of modality and about the

idea that the use of inference to the best explanation is an acceptable method for justifying

the kind of metaphysical existence claims Lewis’s theory asserts. But it would fall beyond

the purview of this dissertation to evaluate Lewis’s use of IBE-style reasoning in support

of modal realism.9 Moreover, I think that on this occasion it is appropriate to sidestep

the issue about whether, in general, Lewis is justified in claiming that there exist possible

worlds. This is because even if Lewis has a plausible case for the general claim that there

exist possible worlds, that would nevertheless fail to address the specific issue of whether

it is acceptable to appeal to his reduction of modality in terms of worlds in order to justify

assertions about what is possible.

My interest shall be, then, on what Lewis has to say about justifying modal assertions.

As it turns out, to whatever extent that it is possible to justify modal assertions under

Lewis’s reduction, this justification accrues in relative isolation from his analysis of modal-

ity in terms of worlds, and has instead to do with some of the ancillary components of

modal realism, viz., the hypothesis that commonsense modal intuitions are largely justified

as they come. Note that most of what I say below is not intended as a criticism of modal

realism, which was not constructed for the purpose of (epistemically) justifying modal

assertions.10 Rather, I am attempting to establish a more general precedent for my reply to

Shapiro, which is to deny that reductions of modality, solely in virtue of their reductive

characters, provide a means for justifying modal assertions.

3.3.2 Lewis and Justifying Modal Assertions

According to an early objection due to Tom Richards, even commonsense beliefs about

modality are largely unjustified under Lewis’s reduction. Given that other possible worlds

are spatiotemporally isolated from the actual world, it stands to reason that no one is

9See (Shalkowski 2010) and (Shalkowski 2012) for criticism of Lewis’s IBE/theoretical-utility justificationfor modal realism.

10Though it does, of course, offer metaphysical justifications or grounds for modal assertions. So it is worthrecalling here that lone instances of ‘justify’ refer to the epistemic sense of the term as described in §2.



capable of “inspecting” other possible worlds in order to find out what holds true in them.

Thus, if being possible genuinely is being true in a possible world, then no one knows

what is merely possible, since no one has knowledge of what transpires in other worlds

(Richards 1975, 109-10). Lewis replies by arguing that since commonsense beliefs about

modality are largely justified, then the truth of his theory shows that knowledge about

what transpires in other worlds is unproblematic in at least some cases. For instance, the

intuition that it is possible for there to be talking donkeys is all the evidence that is required

under Lewis’s theory for asserting that there indeed exist possible worlds (countless worlds,

thinks Lewis) in which there are talking donkeys (Lewis 1986, 110).

Lewis’s response to Richards, then, is to argue that it is possible to have knowledge

about what transpires in other worlds, because already existing and justified modal

knowledge, together with the left-to-right hand readings of (1) and (2), justify some

assertions about what is true in some merely possible worlds. This, however, is not to

contest Richards’s objection, which is that modal knowledge is not acquired by inspecting

merely possible worlds. As Richards observes, if Lewis is right about how knowledge

about possible worlds is acquired—i.e., that this knowledge is acquired, “according to

one’s lights about what is possible”—then possible worlds, “can shed no light on the

question of what is possible” (Richards 1975, 112). Lewis does not contest this, and further,

there does not appear to be any important reason why he must contest this.11 After all, to

claim that possibility can be analyzed in terms of possible worlds, and that what transpires

in other worlds comprises the truth-conditions of modal assertions, does not require a

theory about the extent of possibility—a theory about what is and is not possible. Nor does

such an analysis require a concomitant theory of how anyone comes to learn about what

is and is not possible.12 It is somewhat surprising, then, to see Richards draw the further

conclusion that Lewis’s theory, “does not explain the meaning of ‘possible,’ ” because

Lewis’s worlds shed no light on what is possible (Richards 1975, 112). To borrow from

11Thanks to Michael McKinsey for some useful suggestions here.12See (Cameron 2012) for a more detailed working out of this line of reasoning.



a related observation of (Cameron 2012), Richards appears to run together three distinct

tasks—an account of the extent of possibility, an account of the epistemology of possibility,

and an account of the meaning of the term ‘possible.’ It is perfectly reasonable that a

reductive theory of modality could complete the last of these tasks without completing

either of the first two. (Of course, it is reasonable to wonder what a theory of modality is

good for if it does not help carry out either of the first two tasks—this is perhaps a more

charitable reading of Richards’s criticism.)

The relevance of this discussion to the question of whether reduction assists in the

justification of modal assertions is as follows: By Lewis’s own admission, knowledge about

what transpires in other worlds is derivative of previously established modal knowledge.

Lewis’s reduction of modality to worlds, then, does not do any work justifying modal

assertions. The reduction merely serves as a device for translating modal assertions into

assertions in terms of worlds, and vice versa, and in the process purports to reveal the

truth-conditions for modal assertions. The role of the reduction, then, is to serve simply as

a rewrite rule: Prior confidence that ‘p is true in a possible world’ can be used as evidence

that ‘p is possible;’ and prior confidence that ‘p is possible’ can be used as evidence that ‘p

is true in a possible world.’ But that ‘p is possible just in case p is true in a possible world’

nowhere provides independent support for lone instances of ‘p is possible’ or lone instances

of ‘p is true in a possible world.’

Now, I think it is worth pointing out that there are good reasons for supposing that,

at least in an epistemic sense, Lewis is justified neither in deducing assertions about

worlds from modal assertions, nor in deducing modal assertions from assertions about

worlds.13 This suggests that, if Lewis’s reductive analysis is true, then modal assertions are

fundamentally unjustifiable. This is because once the connection from modal assertions to

13For criticism that one cannot deduce claims about worlds from modal claims, see (Bueno and Shalkowski2000). The criticism that one cannot deduce modal claims from claims about worlds is hinted at in (vanInwagen 1998), but not made explicit. I am confident that the criticism can be made explicit (and persuasive),but since establishing this criticism is more than is required for my purposes, I will have to leave matters asthey stand.



worlds has been severed, no longer can Lewis insist that commonsense modal intuitions

yield reliable insights into what transpires in other worlds, cutting off entirely any access

one might have to other worlds—worlds that still must provide the truth-conditions for

modal assertions, according to Lewis’s analysis. It follows that it is impossible to justify

modal assertions (barring what can be deduced from the actual world together with modal

logic), at least under the assumption that justifying modal assertions presupposes or

otherwise involves confidence that their truth-conditions obtain or are genuinely satisfied.

This is, however, a stronger conclusion than is required for my purposes, as I only wish to

substantiate the claim that reduction on its own does not help justify modal assertions.

Nevertheless it would be incorrect to claim that Lewis says nothing further about

how to justify modal assertions, and it would also be incorrect to claim that Lewis says

nothing further about the extent of possibility. He provides, at least in a preliminary way,

an account of both items, and each involves his principle of recombination. Recall that

this principle states that for any two individuals, there is a possible world containing

qualitatively identical duplicates of those individuals. This principle is what generates or

constructs Lewis’s entire modal universe from his basic mereological resources, thereby

providing him with an account of the extent of possibility. Thus, if p is possible, then p

refers to some state of affairs that can be constructed from Lewis’s basic mereological

resources by repeated application of the principle of recombination. In other words, ‘being

a state of affairs constructed from basic mereological resources by (repeated) application

of the principle of recombination’ is coextensive with, ‘being true in a possible world.’

With respect to the justification of modal assertions, Lewis hypothesizes that common-

sense modal intuitions are largely justified as they come, and further, that commonsense

modal intuitions are consequences of his principle of recombination (or something like it)

(Lewis 1986, 113). There are at least two ways of interpreting what it means to say that com-

monsense modal intuitions are consequences of the principle of recombination. According

to one interpretation, the principle of recombination is evidentially and justificationally



prior to commonsense modal intuitions. To use a mathematical analogy, this would be

to hold that Lewis’s basic mereological resources are like “axioms,” that his principle

of recombination is like a “proof theory,” and that commonsense modal intuitions are

derivable “theorems” that would be entirely unsupported (in both the metaphysical and

epistemic senses of §2) without the former resources. According to a second interpretation,

commonsense modal intuitions abductively justify the principle of recombination, with

this principle providing something like a backwards-looking account of the content of

these intuitions. To again use a mathematical analogy, commonsense modal intuitions

would be equivalent to the kinds of arithmetical and number-theoretic intuitions (e.g., zero

has no predecessor, 2+2=4) that might be thought to abductively justify the Peano axioms.

The Peano axioms, once abductively justified, can be used to derive rather surprising and

highly non-trivial arithmetical assertions, but will generally also turn some of the original

intuitions (e.g., 2+2=4) into rather uninteresting theorems. So too might the principle of

recombination, once abductively justified, be used to “derive” some rather surprising

and highly non-trivial modal “consequences,” in addition to being usable for recapturing

commonsense modal intuitions (e.g., talking donkeys are possible) as “consequences.”

Which interpretation should be attributed to Lewis? Note that Lewis says that the

principle of recombination has uses in “imaginative experiments” in which one tries, “to

think of how duplicates of things already accepted as possible. . . might be arranged to

fit the description of an alleged possibility” (ibid., 113-4).14 That commonsense modal

intuitions, by definition, fall under the province of commonsense, suggests that such

intuitions are perfectly justified as they come. In other words, these intuitions serve to

identify a loose conglomeration of assertions that are already accepted as possible. The

principle of recombination is not required, then, in order to justify these intuitions (even

if it is possible to use the principle of recombination in this way)—commonsense modal

14Lewis acknowledges that, “[f]or more far-fetched possibilities, recombination is less useful” (ibid., 114),presumably because it is unclear just what far-fetched possibilities are recombinations of. Of course, whatcounts as a far-fetched possibility appears to be unhelpfully subjective—perhaps talking donkeys are notfar-fetched for Lewis, but (van Inwagen 1998) seems to think otherwise.



intuitions instead are fodder for the derivation of further modal “consequences.” This

suggests that Lewis would opt for the second interpretation over the first.

But I do not see that, as far as my purposes are concerned, there is much that hangs

on sorting out, under Lewis’s overall theory of modality, the precise relationship between

commonsense modal intuitions and the principle of recombination. According to the first

interpretation, all of the justificatory work is performed by the principle of recombination.

According to the second interpretation, most (if not all) of the justificatory work is per-

formed by commonsense modal intuitions (perhaps with the principle of recombination

helping to extend modal knowledge beyond commonsense modal intuitions). But again,

in neither case does the reduction of modality to worlds serve as anything more than a rule

for translating modal assertions into worldly assertions (which are elliptical for descrip-

tions of complex products of the principle of recombination) and for translating worldly

assertions into modal assertions. So whether Lewis’s principle of recombination provides

an acceptable means for justifying modal assertions is largely immaterial, because the

reductive portion of his theory still serves as a mere rewrite rule, and does no justificatory

work. That work is left entirely to some admixture of commonsense modal intuitions and

applications of the principle of recombination.

Let me reemphasize that I have not attempted to mount either a criticism of Lewis’s

reduction of modality in terms or worlds or a criticism of Lewis’s views about the justifica-

tion of modal assertions. What I have identified is only the idea that the fact that modal

realism is a reductive theory of modality is of no significance whatsoever to modal realism’s

prospects for justifying modal assertions. Nonetheless, this idea does suggest a criticism of

Shapiro’s “epistemic” objections to nominalism, which holds that reduction facilitates a

more tractable modal epistemology.


Reducing Modality 164 Possible Worlds and Functional Roles

3.4 Possible Worlds and Functional Roles

The upshot of the previous section is that although Lewis’s reduction is capable of meta-

physically justifying modal assertions (because it provides truth-conditions for modal

assertions), the reduction itself does nothing at all to epistemically justify modal assertions.

I believe that this observation generalizes to most, if not all reductions of modality—at

least, to most or all reductions that utilize the possible worlds metaphor—because, as I

shall argue, the observation has important structural elements.

Nearly every theory of modality grants credence to the notion that the possible worlds

metaphor is useful for identifying the functional roles that analyses of necessity and

possibility ought to fill.15 Structurally, there is little that distinguishes Lewis’s reduction of

modality from all other accounts of modality that seek to vindicate Kripke-style modal

semantics, including the accounts given by (Armstrong 1989), (Peacocke 1999), (Plantinga

1974), (Rosen 1990), and (Stalnaker 1984). It would seem that Kripke-style modal semantics

constitutes something of a shared or background theory that provides implicit, functional

accounts of necessity and possibility: There is a non-empty domain of ϕs; that p is necessary

means that p is true in all ϕs; that p possible means that p is true in a at least one ϕ. Each

theory then analyzes necessity and possibility in terms of the particular entities they

substitute for ϕ. In the case of (Armstrong 1989), the entities are maximal states of affairs;

for (Plantinga 1974) the entities are maximally consistent sets of propositions. Of course, in

Lewis’s case, the entities are spatiotemporally disconnected universes. What distinguishes

these views, then, is not the structure of their accounts of modality, but rather the particular

metaphysical commitments they take to be required for eliciting the truth-conditions of

modal assertions.

For each theory, then, it is a mere triviality that necessity just is truth in all ϕs and that

possibility just is truth in a ϕ (substituting for ϕ). Before substituting for ϕ, the structure

15Thanks to Eric Hiddleston for suggesting this way of looking at the debate. For a full-blown functionalistaccount of modality, see (Gregory 2006).


Reducing Modality 165 Possible Worlds and Functional Roles

simply describes a functional role—a role that is to be filled by different entities depending

on which theory is adopted. But notice that these general analyses are mere rewrite rules—

that possibility just is truth in a ϕ does not provide independent support for claims like ‘p

is possible’ or for claims like ‘p is true in a ϕ.’ It is true that prior confidence that ‘p is true

in a ϕ’ together with the truth of the general analysis will entail that ‘p is possible.’ But

that would not be an explanation for why one is justified in holding that ‘p is true in a ϕ,’

which almost certainly would be in doubt if one were in doubt about whether p is possible.

What this means is that, in general, reduction is of little use when one’s interest lie in

justifying modal assertions. Reduction can be used to translate an already justified modal

assertion into a justified assertion about the domain of ϕs, and it can be used to translate

an already justified assertion about the domain of ϕs into a justified modal assertion. But

reduction cannot be used to provide initial justification to any modal assertion or to any

assertion about what transpires within the domain of ϕs. To say otherwise would be to

confuse an account of the truth-conditions for modal assertion for an account of what

justifies modal assertions.

In the next section I shall argue that the set-theoretic reduction of the logical modalities,

like Lewis’s reduction of the metaphysical modalities, provides another example of the

general idea that reduction does not serve to justify modal assertions. If that is right, then

since the truth of set theory is in at least as much doubt as the truth of the modal assertions

modal nominalists make, it follows that it would be incorrect for Shapiro to insist that

someone in his position is better able to justify the modal assertions nominalists make.

If these assertions are unjustified for modal nominalists, they are no less unjustified for

set-theoretic reductionists. Moreover, since the truth of set theory is in doubt, Shapiro

cannot establish with any kind of certainty that the reductive truth conditions for logical

possibility assertions are in fact realized.


Reducing Modality 166 Lessons for Shapiro and Set Theory

3.5 Lessons for Shapiro and Set Theory

According to Shapiro, set theory provides the metaphysical resources or grounds for sup-

porting assertions about the logical modalities—if not in general, then at least in application

to logical possibility and necessity claims about the truth or not of mathematical theories

and the existence or not of mathematical objects. The idea here is that the truth-conditions

for such claims are supplied by facts about what kinds of set-theoretic constructions can

be modeled in set theory. The notion of logical possibility is linked to being true in a model;

logical necessity to being true in all models. To paraphrase Shapiro, the idea is that the

set-theoretic universe is so rich and expansive that it provides the materials to construct or

model every possible mathematical object. After describing the basic working components

of the set-theoretic reduction, I show that the analyses of logical necessity and possibility in

terms of models serve as mere rewrite rules and confer no justification upon lone assertions

about what is logically possible or necessary. Thus, the set-theoretic reduction is not in

itself a useful implement for justifying these assertions. For the remainder of this section

the reader should assume, unless told otherwise, that the modalities under discussion are

the logical modalities.

3.5.1 The Set-Theoretic Reduction

The model-theoretic treatment of the logical modalities has quite a tumultuous history—

C.I. Lewis’s pioneering work in modal logic precipitated Quine’s famous criticism16 and

set numerous philosophers and logicians to work attempting to produce a viable, non-

modal interpretation for the semantics of modal logic. I will not attempt an overview of

the dispute between Quine and the modal logicians.17 The upshot is that Kripke-style

possible-worlds semantics have become the standard formal vehicles through which to

16“There are logicians, myself among them, to whom the ideas of modal logic (e.g. Lewis’s) are notintuitively clear until explained in non-modal terms” (Quine 1947, 43).

17The interested reader is encouraged to consult (Marcus 1993b), (Ballarin 2004), and (Ballarin 2005) fordiscussion and references.


Reducing Modality 167 The Set-Theoretic Reduction

explicate the logical modalities.18

For modal propositional logic,19 one defines a model structure as an ordered triple

(G,K,R) in which K is a non-empty set, G is a member of K, and R is a relation on K. A

model is a binary function ϕ(p,HεK) (for atomic formulae p) that produces as values T or F.

Non-atomic formulae are assigned values inductively. � and ♦ are sentential operators

that obey the following equivalences:

5. ϕ(�p,H) = T ≡ ϕ(p,H ′) = T for every H ′εK : HRH ′; otherwise ϕ(�p,H) = F .

6. ϕ(♦p,H) = T ≡ ϕ(p,H ′) = T for some H ′εK : HRH ′; otherwise ϕ(♦p,H) = F .

Intuitively, K is a set of possible worlds, G is the actual world, and R is an accessibility

relation on K (a number of systems of modal logic can be differentiated on the basis of

whether this relation is transitive, symmetric, etc.). Here ‘accessibility’ can be interpreted

as ‘possible relative to’—thus to say that GRH is to say that H is a possible world relative

to the actual world. A model, then, determines what propositions are true in what possible

worlds. (5) and (6) state, respectively, that a sentence is necessarily true just in case it is

true in every possible world (and not necessarily true otherwise); and that a sentence is

possibly true just in case it is true in at least one possible world (and not possibly true

otherwise).20 Of course, as many expositors note, this intuitive picture is offered merely

for heuristic purposes—in describing modal semantics readers are often warned against

supposing that one is literally quantifying over things called ‘possible worlds.’

How can the model-theoretic treatment of the logical modalities serve as a reduction?

First I need to clear up an ambiguity between two different senses or uses of the term

‘model.’ In Kripke-style semantics, a model is defined as a function from propositions and

worlds to truth-values. Below I will refer to this use of ‘model’ by the term ‘modal-model.’

18Similar formalisms were antecedently developed by Jaakko Hintikka and Stig Kanger.19What follows is drawn from Kripke (1963).20For modal predicate logic, the formalism must be expanded; each ‘world’ must be assigned a domain of

objects and these objects must be assigned predicates. A tremendous amount of ink has been spilled overwhether this domain is allowed to vary in size and makeup from world to world.



This modal use of the term ‘model’ is distinct from the ordinary, non-modal use of the term,

in which a model is merely an interpretation of the non-logical vocabulary of some sentence

or set of sentences. I will use the non-modified ‘model’ to refer to this ordinary use of the

term. Note that Kripke-style semantics invokes both usages. Each of Kripke’s “worlds” can

be thought of as a model (in the ordinary sense); meanwhile, each modal-model includes a

set of models.21 I mentioned previously that the logical notions of possibility and necessity

can be linked to being true in a model and being true in every model (respectively). These

associations should now appear ambiguous—is logical possibility to be analyzed as truth

in at least one modal-model, or simply as truth in at least one model? The answer is the latter.

The set-theoretic reductionist’s gambit is that the universe of sets contains the resources

to construct or model all of the logical possibilities. Thus, the set-theoretic hierarchy

constitutes the domain for the alethic modal-model.

Note that to be effective as a reduction of logical modality, set theory must furnish the

truth-conditions for claims of logical possibility and necessity—otherwise set theory at best

serves as a mere representational device. However, there is a certain awkwardness associ-

ated with insisting that set theory provides the truth-conditions for all logical possibility

claims. Presumably it is logically possible that Larry had become a fishmonger rather than

a philosopher, but it would be implausible to argue that the content of the claim that ‘it is

logically possible that Larry is a fishmonger’ is that there exists a pure set of a certain sort.

Thus, it is implausible to claim that set-theory provides reductive truth-conditions for all

logical possibility assertions. (Though it could be said that set theory provides the means

for modeling or representing all logical possibilities—e.g., as a means for making it evident

that ‘Larry is a fishmonger’ does not lead to a logical contradiction.) One the other hand,

there seems to be little awkwardness associated with the claim that set theory provides

truth-conditions for logical possibility claims about the existence of mathematical objects

and for logical possibility claims about the existence of models of mathematical theories.

21For a similar observation, see (Ballarin 2005, 284).



Indeed, Shapiro seems confident that set theory is comprehensive in this way (1997, 136).22

3.5.2 Justifying Modal Assertions Under the Set-Theoretic Reduction

Scott Shalkowski (2004) has advanced something like a set-theoretic analogue of the kind

of epistemic objection Richards’s raises against Lewis’s reduction. Recall that Richards’s

objection to Lewis is essentially that, if knowledge about what is metaphysically possible

requires some prior means of inspecting Lewis’s possible worlds, then since no one is

capable of making such inspections, it follows that no one knows very much about what is

metaphysically possible. In application to set theory, Richards’s objection would be that if

knowledge about what is logically possible requires some prior means of inspecting sets

and models, then if no one is capable of making such inspections, then no one knows very

much about what is logically possible. What Shalkowski purports to show is that the only

means anyone has for inspecting sets and models is via prior intuitions about logic and the

logical modalities.

According to Shalkowski, “[t]he epistemology of model theory relies on what mathe-

maticians and logicians have antecedently taken to be valid inference as well as what they

have taken to be initially plausible candidates for axioms of set theory” (ibid., 70-1). “For

instance,” he continues somewhat later,

. . . it is assumed by classical logicians that models obey the standard principlesof classical logic: Noncontradiction and Excluded Middle. When we arrive at

22One important obstacle to the idea that set theory provides a bona fide modal-model (even for logicalpossibility claims about mathematical theories) is that there are multiple non-isomorphic set theories.Presumably more than just one of these theories is consistent. But if that is the case, then which set theory isit that produces the alethic modal-model for logical possibility? Or should logical possibility be analyzedas what can be modeled or constructed in any consistent set theory? This multiplies the difficulties on twofronts: First, in order to be genuinely reductive, all of these consistent set theories must be true. Since theyare non-isomorphic theories, this new reduction is now committed to the existence of multiple set-theoretichierarchies, sensu the full-blooded or plenitudinous platonism of (Balaguer 1998). Second, the notion ofconsistency as it applies to the various set theories cannot be treated set- or model-theoretically. The troubleis that the model-theoretic account of the logical modalities is theory-relative. It provides a means for decidingthe modal profiles of sentences only under the assumption of some background set theory—but, importantly, itdoes not appear to have anything to say about the modal profile of this background theory. One could attempt tomove to a meta-set theory in order to capture something like a model-theoretic reduction for all of thesemodal-models. But I do not think this would be a feasible option except under the assumption of a universaldomain—a highly problematic assumption and one that I do not have space to evaluate in any detail.



surprising results like the Lowenheim-Skolem theorems, our use of assumedlogical facts do all the work in telling us about the number and nature ofmodels. Whether the process is one of construction or discovery, the logicaldog is wagging the model-theoretic tail. . . Those with nonstandard ideas aboutlogical principles will, understandably, have nonstandard ideas about theexistence and representative capacities of the set of models. (ibid., 71)

Shalkowski’s moral is relatively clear, at least from an epistemological point of view:

Insight into the nature of the set-theoretic hierarchy is entirely dependent on prior assump-

tions about logic and logical possibility. Why does the set-theoretic hierarchy contain the

particular members that it does? Well, because the membership of this hierarchy can be

characterized by a set of jointly logically possible axioms together with some or other

account of logical consequence. But what makes those axioms jointly logically possible?

It would clearly be circular to insist at this point that the axioms of set theory, e.g. ZFC,

are jointly logically possible because they have a model. What this suggests is that the

set-theoretic reduction itself does not assist in the justification of modal assertions (or of

assertions about set existence).

Shalkowski, like Richards, opts for a somewhat stronger conclusion, viz., that it is

not possible to non-circularly justify the truth of the set-theoretic analyses of the logical

modalities, because no one can guarantee that the logical principles used to construct the

set-theoretic hierarchy are genuinely in accord with the facts of logical possibility (ibid.,

72). I suspect that this charge of circularity makes something like the mistake Richards

makes—to conflate an identification of the truth-conditions for modal assertions with

an account of the extent and epistemology of modality—but I shall not press the issue.

I am content to note that the set-theoretic reduction itself confers no justification upon

modal assertions, and I think this point can be sustained without accusing the set-theoretic

reduction of engendering circularity. The explanation for why the set-theoretic reduction

itself confers no justification upon modal assertions is simply that the set-theoretic analyses

of the logical modalities, like Lewis’s analyses of the metaphysical modalities, serve only

as rewrite rules. The only difference is that in this case the analyses are rewrite rules



for translating modal assertions into set-theoretic assertions, and vice versa. Thus, that

possibility can be analyzed as truth in a model does nothing to justify lone modal assertions,

or lone assertions about what models exist.

What is the significance of these developments for Shapiro’s criticism of nominalism?

Well, if it is possible to justify some set existence claims, then it is possible to use the right-

to-left readings of (5) and (6) to justify some modal assertions. For this reason, Shapiro

would have a decisive advantage over the modal nominalist if he were in a position

to justify set existence claims—in particular, if he were able to justify the set-theoretic

correlates of the modal assertions modal nominalists make. Therefore it is worth restating

the case for why Shapiro is not in a position to justify set existence claims—a case made in

some detail in the previous chapter. On Shapiro’s view, the positions of ante rem structures

are themselves bona fide mathematical objects, which means the existence or not of sets is

a consequence of the existence or not of the ante rem set-theoretic structure. That such a

structure exists for Shapiro follows from the coherence—a modal primitive—of set theory.23

What I argued in the previous chapter was that the inference from the coherence of set

theory to the existence of the set-theoretic structure (and hence, the existence of sets) is

entirely unjustified. Coherence does not imply existence, even for structures. Thus, Shapiro

is not himself in a position to justify set existence claims, even if he is in a position to justify

claims about what mathematical theories are coherent. It follows that Shapiro does not

have any kind of advantage over the modal nominalist in virtue of his insight into the

set-theoretic hierarchy.

Is Shapiro in a position to justify coherence assertions? And might he claim some kind

of advantage over modal nominalists concerning the justification of modal assertions, by

way of any facility he might have for justifying coherence assertions? I think that both

prospects must be rejected. It must be recalled that coherence itself is a primitive notion,

and given its indistinctness from second-order logical possibility, it would seem to be a

23The general idea here is codified as an axiom of Shapiro’s structuralism—The Coherence axiom, whichstates that coherent formulae of second-order languages are satisfied by actually existing structures.



kind of modal primitive.24 Thus, the same metaphysical and epistemic concerns that apply

to modal nominalists, qua modal primitivists, also apply to Shapiro, qua modal primitivist

(re coherence). Shapiro is therefore as much in the dark about coherence, metaphysically

and epistemically, as modal nominalists are about primitive modality.

Shapiro, of course, insists that it is a presupposition of mathematics that set theory is

coherent (1997, 136), which he appropriates in support of the idea that he is epistemically

justified in making coherence assertions (despite apparently lacking metaphysical justifica-

tions for coherence assertions). But given the indistinctness of coherence and second-order

logical possibility, this presupposition can be appropriated by modal nominalists in sup-

port of the claim that the axioms of set theory are jointly logically possible. Thus, modal

nominalists would appear to have an equally reliable epistemic justification for logical

possibility claims (despite their nevertheless having no metaphysical justification for these

claims). Shapiro’s insistence that it is a presupposition of mathematics that set theory is

coherent, then, concedes quite a lot to the modal nominalist. This would be, in Lewisian

lingo, to identify the coherence or logical possibility of set theory as a commonsense

intuition about the logical modalities.25 Such commonsense intuitions are justified for

all, if they are justified for anyone. Modal nominalists are not blind to the presupposi-

tions of mathematics, whatever these might be, and it would be special pleading were

Shapiro or anyone else to insist otherwise. Whether it is a commonsense modal intuition

that set theory is coherent is debatable, but nothing here turns on how this debate gets

resolved: If the coherence of set theory is a commonsense modal intuition, then Shapiro

24And given this, it is reasonable to wonder whether the set-theoretic reduction of the logical modalitieswould be genuinely reductive under Shapiro’s structuralism—Shapiro is in effect making the dual proposalthat logical possibility is reduced to set existence, and set existence is reduced to coherence. This chain isvery tight, even if it does not double back upon itself. I find it somewhat curious that Shapiro recognizesthat it would beg the question to reduce coherence to set theory (1997, 135), while at the same time failing torecognize that the inherent affinities between coherence and logical possibility suggest that it would similarlybe question-begging to reduce logical possibility to set theory. Of course, even if these notions were distinctenough to make it plausible that logical possibility is reducible to set theory, while coherence is not, Shapirois nonetheless stuck with the modal primitive of coherence.

25Cf. Shapiro: “Structuralists accept this presupposition and make use of it like everyone else, and we arein no better (and no worse) of a position to justify it” (ibid., 136).



and the modal nominalists are on an equal, positive footing concerning their ability to

justify coherence and logical possibility claims about set theory; if the coherence of set

theory is not a commonsense modal intuition, then Shapiro and the modal nominalists

are on an equal, negative footing concerning their ability to justify coherence and logical

possibility claims about set theory. Either way, Shapiro’s elaborate “epistemic” criticism of

modal nominalism does not show that the modal nominalist (qua modal primitivist) is in

a uniquely disparaging position regarding the justifiability of modal assertions, because

not even Shapiro (qua set-theoretic reductionist) is in a better position to justify modal

assertions. In any case, it would certainly be false to say that the set-theoretic reduction

itself does any work for Shapiro justifying modal assertions—that work seems entirely

derivative of what is presupposed in mathematics.

My basic point is this: A reduction of logical possibility to set theory does no epistemic-

justificatory work on its own. If Shapiro is correct to insist that commonsense modal

intuitions are too vague to support the modal assertions modal nominalists make (ibid.,

232), then the same must be said about the relationship between commonsense intuitions

and the coherence assertions Shapiro makes. Since coherence assertions provide the

metaphysical and epistemological grounds for his set existence claims, and these existence

claims in turn provide the truth-conditions for modal assertions (under the set-theoretic

reduction), then any disconnect between commonsense intuitions and coherence assertions

is ultimately parasitic on both set existence claims and modal assertions. If Shapiro cannot

justify coherence assertions, then he can justify neither set existence claims nor modal

assertions.

Of course, in all probability Shapiro is correct to say that an understanding of the modal

notions is mediated by mathematics and by set theory (ibid.)—regardless of whether ‘un-

derstanding the modal notions’ here means the ability to metaphysically or epistemically

justify modal assertions. At least as far as the epistemic justification of modal assertions

goes, it is also likely that the best evidence available for or against the coherence or logical


Reducing Modality 174 Living With Primitive Modality

possibility of a mathematical theory comes from the theory’s fruitful employment in math-

ematics (perhaps in conjunction with extensive, failed efforts to prove the inconsistency of

the theory). But Shapiro is only entitled to say these things without the prejudicial reading

according to which an understanding of the modal notions is only possible when mediated

by a platonistic interpretation of mathematics and set theory. Although the coherence or

logical possibility of set theory and other mathematical theories may be presuppositions

of mathematics, it is no presupposition of mathematics that sets and other mathematical

objects exist.26 It would be an error to claim that it is a violation of nominalism to use

established mathematics to inform debate about coherence and modality.

Modal nominalists, qua modal primitivists, might be unable to metaphysically justify

or ground modal assertions in the sense that they are unable to articulate the truth-

conditions for these assertions. But as far as modal assertions about mathematics are

concerned, they would appear to be able to epistemically justify modal assertions in

the sense that mathematical practice provides a reason to believe that these assertions are

true. What extinguishes Shapiro’s criticism, then, is (a) that Shapiro similarly cannot

articulate the truth-conditions for coherence assertions—these assertions for him remain

metaphysically unjustified—and (b) that the only clear epistemic justification he has for

coherence assertions comes from an appeal to mathematical practice. Shapiro and the

modal nominalists, then, are essentially in the same boat, hence it is incorrect to accuse

modal nominalism of raising particularly vexing questions about the metaphysical and

epistemic justification of modal assertions.

3.6 Living With Primitive Modality

I suspect that nominalists are driven to accept primitivism about modality because no

reduction on record avoids commitment to some kind of abstract ontology. A basic

presupposition of the debate between modal nominalists and platonists like Shapiro is that

primitive modality is not ontologically committing. For instance, Mark Balaguer insists

26For more here, see chapters 4 and 5, passim.



that primitive modality is nominalistically acceptable because, “it is entirely obvious that

[modal primitives are not] defined in terms of abstract objects, because [they do not] have

any definition at all” (1998, 71). But is it true that modal primitivism avoids commitment

to any kind of abstract ontology?

Consider again the functional understanding of possibility and necessity from §4:

Kripke semantics provides an attractive structure or framework with which to construct

an account of modality: There is some domain of ϕs; necessity is truth in all ϕs; possibility

is truth in a ϕ. Nominalist modal primitivists, just like most other modal theorists, share

an interest in using this kind of structure in order to vindicate the inferential patterns

governing the modal operators. How is the primitivist position to be understood in

reference to this idea? Is modal primitivism best understood as denying that there is

anything that could fill the ϕ role? Alternatively, is modal primitivism best understood as

the bare claim that something fills the ϕ role without any further comment about what that

something is? Or is modal primitivism to be understood in some other way entirely?27

To insist that modal primitivism commits one to denying that anything fills the ϕ role is

tantamount to rejecting the idea that modal assertions have truth-conditions and would

imply either that modal truths do not require truthmakers or that modal discourse is not

truth-apt (McLeod 2001, 27). I suspect that neither implication is desirable. The latter

implication—that modal discourse is not truth-apt—is clearly in conflict with Chihara’s and

Hellman’s motivations to vindicate the claim that mathematics comprises a body of truths.

For instance, if Hellman is correct that the goal of mathematical theorizing is to describe

structural possibilities, then realism in truth-value for mathematical assertions becomes,

under modal structuralism, realism in truth-value for logical possibility assertions.

But alternatively, to insist that modal primitivism comes to the bare assertion that

something fills the ϕ role is potentially inconsistent with nominalism. Without further

specification about just what this something is, modal nominalists can hardly be confident

27Thanks to Eric Hiddleston for helping me to appreciate the significance of these questions.



that it is nominalistically acceptable. Consider the collection of all nominalistically accept-

able entities in the actual universe. Is is doubtful that this collection possess the requisite

structure to fill the ϕ role. Further, it is implausible that there are enough nominalistically

acceptable entities to even serve as proxies for all genuinely possible assertions. Some

kind of increase in ontology appears required in order to account for every possibility.

Thus it is not clear that primitive modality is as ontologically innocent as modal nomi-

nalists presume. For instance, Chihara’s modal primitives—ways the world might have

been—sound much like uninstantiated properties.28 Whether it is possible to provide a

nominalistically acceptable theory of modality therefore appears in doubt and remains an

open question—one that I cannot settle at the moment.

Independent of the question of whether the view is nominalistically acceptable, it is still

worth considering whether modal primitivism is an attractive account of modality. Perhaps

modal primitivism presents the most ontologically austere account of modality—perhaps

it is committed to the smallest ontology necessary for filling the ϕ roles.29 The primitivist

cannot state exactly what this ontology is, but she can be confident that it falls well short

of Lewis’s full-blooded possible worlds. Perhaps she can get away with a pure form of

structuralism about modality in which she is committed to only to what Shapiro would

describe as an ante rem possible worlds structure.30 How does the primitivist justify the

bare claim that there is something (rather than nothing) that plays the ϕ role? According

to Shalkowski, it is not altogether difficulty to justify this claim:

Whether we modalize, then, rests on whether it is conducive to a general frame-work of understanding. For those who resist modal skepticism the activity ofmodalizing should be relinquished only at the greatest cost, since logical truthand elementary arithmetic seem to require modality. . . If we disabuse ourselvesof the thought that modal knowledge must, somehow, be certain knowledgeor that it must, at least, be knowledge more certain than empirical knowledge,then we can allow that primitive modality is required by a comprehensive

28Thanks to Eric Hiddleston for this observation.29(Gregory 2006) argues that the set-theoretic hierarchy is the least controversial ontological posit that is

capable of filling the ϕ roles.30(Sider 2002) advances a similar proposal.



and satisfactory theory which enables our understanding of the world and ourplace in it. (1996, 389)

Shalkowski suggests that there is sufficient holistic evidence for supposing that something

plays the ϕ role.31 Of course, it is open to Lewis and Shapiro to defend their reductions

on holistic grounds. But the analyses that facilitate these reductions do no extra work

justifying modal assertions. And their added metaphysical hypotheses—the existence of

possible worlds and the existence of sets, respectively—represent increased risks without

any consequent gains. Modal primitivism is weaker, and hence more likely to be true. But

crucially, modal primitivism does not sacrifice any modal content that is actually pivotal

for justifying modal assertions. Assertions about the existence of the relevant content (for

justifying that the truth-conditions for modal assertions obtain) appear to be no easier to

justify than modal assertions in the first place.

There are provisional grounds, then, for thinking both that modal primitivism is a

defensible position and also that it is preferable to theories that make substantial meta-

physical hypothesis about the content of modal assertions. Nevertheless it is not clear that

modal primitivism is genuinely nominalistic—primitive modal operators and primitive

modal properties must apply to something, and there is no guarantee that this something is

something a nominalist would be comfortable with. However, I think the modal nominalist

can comfortably dismiss Shapiro’s criticism that the modal nominalist’s use of modality

presents uniquely challenging metaphysical and epistemological difficulties: Any problem

the modal nominalist (qua modal primitivist) faces a propos of justifying modal assertions is

likely to be faced by any theorist interested in producing an account of modality. Crucially,

since the set-theoretic reduction does no work justifying modal assertions, Shapiro fails to

show that the set-theoretic reductionist is in a superior position with respect to justifying

these assertions.

31For Shalkowski this comes to the rather imprecise claim that, “some form of modality is a basic,irreducible—primitive—feature of reality” (ibid., 388).


178

Part 2: Naturalism

Neurath has likened science to a boat which, if we are to rebuild it, we must rebuild plank by plank

while staying afloat in it. The philosopher and the scientist are in the same boat.

—W.V. Quine

. . . if we are sailors rebuilding our ship plank by plank on the open sea, then I know of some cargo

we might want to jettison.

—George Boolos

Naturalism in Mathematics

It is often said that the majority of philosophy from antiquity to the present can be

described as a sequence of reactions to Plato. On a less grand scale—but with greater

conviction—it can be said the last 60 years of metaphysics and logic can be described as a

sequence of reactions to Quine. This point holds true for the philosophy of mathematics,

perhaps more so than for any other subdiscipline of analytic philosophy. It is impossible

to deny Quine’s influence in contemporary philosophy of mathematics. His views on

science and ontology pervade the discipline, even if many scholars ultimately reject

them. His greatest legacy to the field is arguably his development and endorsement

of the philosophical orientation of naturalism. Quine offers a picture of philosophical

methodology according to which the methods of empirical science can and should restrict

philosophy in important ways. Philosophers must abandon the “Cartesian Dream” of

lusting after a priori certainty in any area of inquiry. Beliefs of any kind are subject to

confirmation or disconfirmation on the basis of how well they comport with the theory

that does the best job of making sense of one’s sensory experiences. Whether a theory does

the “best job” is to be determined by subjecting it to scientific criteria of theory selection


179

(fecundity, familiarity, scope, simplicity, and of course accordance with observation). In

sum, Quine calls for the abandonment of inherently philosophical standards of warrant and

instead defers to scientific standards of warrant.

The influence of Quine’s naturalism in the philosophy of mathematics is partly respon-

sible for focusing the discipline away from foundational questions and refocusing it on

questions about the practice of professional mathematics. According to the dominant view,

the practice of mathematics is something to be explained as opposed to something in need

of justification. Thus the task of philosophy of mathematics is to explain the proper role of

mathematics as a part of the overall scientific enterprise. This could involve determining

what conceptual, logical, and ontological resources mathematics requires. To use one of

Quine’s metaphors, the philosopher of mathematics is adrift on the Ship of Neurath, and

must determine what kind of a hull enables her to erect the mast of mathematics.

For better or worse, Quine viewed mathematics as a mere handmaiden of empirical

science. This is a view that many naturalists in the philosophy of mathematics reject.

Contra Quine, mathematics is a successful, freestanding discipline, and does not require

the approval and use of the empirical-scientific community to count as an intellectually

worthy pursuit. Nevertheless, as far as influential, well-developed naturalistic accounts of

mathematics go, Quine’s was to stand alone for some 40 years. Only in the past few decades

have new views come out which provide an overall picture of the relationship between

mathematics and the scientific community at-large. In this section of the dissertation, I

reflect upon the challenges that the two most developed post-Quinean naturalists pose

to modal nominalism: John Burgess’s naturalistic objections to modal nominalism and

some objections derived from Penelope Maddy’s “Second Philosophy.” Burgess’s views

are taken up in chapter four; Maddy’s are discussed in chapter five. In essence, Burgess

argues that modal nominalism works against science and mathematics, whereas Maddy

can be understood as arguing that modal nominalism does not work with science and

mathematics. My principal ambition is to show that modal nominalism is consistent with


180

the naturalist orientation in philosophy in the sense that it does not produce any kind of

conflict with the practice of mathematics.


181

Chapter 4

Reflections on Burgess

4.1 Introduction

In this chapter I examine the considerations that have motivated John Burgess to reject

nominalist and modal nominalist accounts of mathematics.1 Burgess has on numerous

occasions advanced the complaint that nominalist theories of mathematics, including

the modal nominalist theories defended in this dissertation, are unscientific, which he

takes to be a reason to believe that nominalism (modal or otherwise) and naturalism

are incompatible doctrines. Although never explicitly laid out in his work, his various

criticisms of nominalist reinterpretations of mathematics fit a pattern of reasoning that I call

Burgess’s “Master Argument” against nominalism. Note that Burgess’s arguments tend to

be targeted at nominalist theories of mathematics in general, and not solely at those theories

that fall under the modal nominalist heading. Further, I think it is possible to respond

to Burgess’s arguments on equally general terms. Thus, the defense I offer for modal

nominalism can be appropriated by other kinds of nominalist theories of mathematics.

For these reasons I will only use the term ‘modal nominalism’ when do so is dialectically

important.

In section two I provide an exposition of the Master Argument and of how it arises out

of Burgess’s various criticisms of nominalism. The basic idea is that there seems to be no

direct scientific evidence in support of the various nominalist accounts of mathematics,

1Most of Burgess’s positive views about science and scientific methodology appear in work he hasco-authored with Gideon Rosen. In the citations below I have taken pains to indicate when the cited textis Burgess’s alone, and when the cited text is a joint work with Rosen. I do not attach much importance towhether I am successful in describing the views of Burgess or whether I only succeed in characterizing aview he and Rosen have in common. The point remains that the corpus of Burgess’s work (including hiscollaborations with Rosen) contains sustained criticisms of nominalism and modal nominalism that deserveexamination. Nevertheless I should call the reader’s attention to the fact that Burgess has indicated in printwhere his views depart from Rosen’s (in his introduction to (Burgess 2008a, 3-5)). As best as I can tell, noneof the views I attribute to him are listed among these departures.


Reflections on Burgess 182 Introduction

including the modal nominalist accounts I aim to defend. This allegedly suffices to

demonstrate that nominalism, including modal nominalism, is unscientific, and hence must

be rejected by naturalists.

Section three examines several of the extant responses to Burgess’s criticism of nom-

inalism. I consider two replies due to Charles Chihara. One of these replies holds that

there in fact does exist scientific and mathematical evidence in support of Constructibility

Theory. I argue that the kind of evidence Chihara provides should be largely unmoving

to Burgess’s naturalist. The second of these replies holds that withholding belief in the

truth simpliciter of mathematical theories is consistent with scientific and mathematical

methodology. I endorse this reply, later funneling it into an ad hominem against Burgess

in the closing section. The most popular response to the various instances of Burgess’s

Master Argument hold that each instance targets a characterization of nominalism that no

nominalist has ever endorsed. That is, nominalists have argued that all of the examples of

the Master Argument that Burgess has actually presented are unsound in that they depend

on mischaracterizations of the rationale behind nominalist philosophies of mathematics.

I argue that this reply, which I call the “False Dilemma” reply, is ultimately insufficient,

because Burgess can adopt the following response: First, the nominalist must supplant

Burgess’s mischaracterizations with clear and identifiable aims for her account of mathe-

matics. Second, she must show that these clear and identifiable aims are not open to some

further manifestation of Burgess’s Master Argument. On behalf of Burgess I argue that

even if a nominalist were to be successful in meeting the first condition, she would be

unlikely to succeed in meeting the second.

I nevertheless contend that Burgess’s criticism is seriously flawed. In section four

I argue that the Master Argument depends on an inadequately articulated position on

what it takes for a statement to be naturalistically unacceptable. Burgess’s evidence that

nominalism is unscientific amounts to little more than promulgating his suspicions that no

scientific journal would ever publish a work on nominalism. His use of expressions such



as ‘p is unscientific’ is equivocal between ‘p is not endorsed by the scientific community or

by scientific methodology,’ and ‘¬p is endorsed by the scientific community or by scientific

methodology.’ I argue that he succeeds only in showing that nominalism is unscientific in

the former sense, and that the naturalist has no prima facie reason to reject claims that are

unscientific only in this sense. Thus, Burgess does not show that nominalism, including

modal nominalism, is incompatible with his naturalism.

I close the chapter in section five with an ad hominem against Burgess’s own favored

view, moderate realism. The moderate realist is a platonist who adopts a literalist reading

of mathematical language. Since no scientific journal would ever publish a work on pla-

tonism, Burgess’s moderate realism is open to the same kind of criticism he levies against

nominalism. Hence, if my objections from section four are erroneous and nominalism

is indeed unscientific (in the damaging sense of the term), then moderate platonism is

similarly unscientific (also in the damaging sense of the term). A key observation here is

that mathematical practice is largely indifferent to matters of interpretation—ontological

presuppositions are not built into the discipline.

Before beginning in earnest, a brief recapitulation is in order of the modal nominalist

strategies.2 In general terms it is useful to distinguish between two sorts of projects—

reinterpretative projects, on the one hand, and reconstructionist projects on the other.

Reinterpretative projects involve constructing novel interpretations of mathematical lan-

guages. Thus reinterpretations constitute a revision to the semantics of mathematical

languages, and need not alter the surface grammar of mathematical assertions. Recon-

structionist projects advocate the ground-up recreation of mathematical theories using

nominalistically kosher primitives. Such projects revise the surface syntax of mathematical

assertions and so offer a more dramatic contrast to standard mathematical languages. An

important point to recognize is that advocates of either sort of project are not committed to

advertising reinterpretation or reconstruction as kinds of activities in which mathemati-

2See the first chapter for a more thorough discussion.



cians must engage. Both can be used as tools for advancing the philosophical understanding

of mathematics. Burgess is not always attentive to these distinctions; I implore the reader

to keep them in mind.

Chihara’s Constructibility Theory replaces existence assertions in mathematical lan-

guages with assertions about the constructibility of open-sentence tokens of various sorts,

his goal being to capture a nominalistic type-theoretic framework. Constructibility Theory

is not offered as an account of the meaning of ordinary mathematical assertions, but instead

as a device for showing that it is possible to engage in mathematical reasoning without quan-

tifying over mathematical objects. This involves the (at least in principle) reconstruction (or

recreation) of the relevant bits of mathematics in Constructibility Theory. Nevertheless

Chihara does not advocate a revision to the practice of mathematics. Constructibility

Theory is designed principally to rebut the various indispensability arguments.

Hellman’s Modal Structuralism constitutes a nominalized version of mathematical

structuralism that postulates the primitive logical possibility of the existence of models of

mathematical theories (as opposed to the actual existence of such models), in addition to

the necessitation of conditionals of the form AX → p, where p is a theorem of the theory

characterized by axioms AX . Hellman advocates Modal Structuralism as an attractive

interpretation of mathematical languages, that is, as a novel reading of ordinary mathemati-

cal language fit to compete with the literalist or realist reading favored by platonists. As in

Chihara’s case, Hellman does not advocate any revision to the practice of mathematics.

Field’s fictionalist account of mathematics holds that mathematical language is to be

interpreted literally. However, he does not assume that any mathematical objects exist,

and hence regards all mathematical existence assertions as false. Field maintains that

mathematical knowledge is a genus of logical knowledge that comes in two basic species:

knowledge of the consistency of mathematical theories (where consistency is taken to be

a primitive logical notion) and knowledge of the consequences of mathematical theories.

Neither reinterpretive nor reconstructive maneuvers are implemented in Field’s account


Reflections on Burgess 185 The Master Argument

of mathematics.3 Consequently, Field does not advocate a revision to the practice of

mathematics.

Each of these philosophies of mathematics purport to advance the philosophical under-

standing of mathematics in various ways. Chihara, by showing that it is possible to engage

in mathematical reasoning without quantifying over mathematical objects. Hellman, by

showing that the platonist’s interpretation of mathematical language can be dispensed

with in favor of a modal-structural interpretation. And Field, by showing that mathemati-

cal theories need not be thought of as expressing truths about mathematical objects. How

are these projects to be evaluated? Burgess’s answer is that each view must answer to the

evidential standards of natural science.

4.2 The Master Argument Against Nominalism

Burgess’s criticism of nominalism is intended to unveil nominalistic philosophies of

mathematics as unscientific, and hence as inconsistent with the naturalistic outlook in

philosophy. The general argumentative strategy he uses, which I am here calling his

“Master Argument” against nominalism, works in the following way. First, Burgess

advances a thesis about the motivation for nominalist reinterpretation. Next, he offers

evidence that the cited motivation is not captured by any evident principle of scientific

methodology. From this he infers that the nominalist’s credo is un- or anti-scientific. In

Burgess’s eyes this suffices to expose nominalism as inconsistent with naturalism, and in

turn, betrays nominalism as a misguided philosophical position.

Burgess is a scientific naturalist, i.e., he believes that natural science is the only reliable

tool for obtaining knowledge about the world, and that matters of science are only to be

criticized from within. Thus he denies that the provenance of philosophy qualifies it as a

provincial authority when it comes to matters of scientific importance. But ‘naturalism’ is

a term much bandied about in philosophy; nearly every philosopher that calls herself a

3Field champions a reconstructionist program in physical theory, to be sure, but he does not do this in hisaccount of pure mathematics.


Reflections on Burgess 186 Burgess’s Naturalism

‘naturalist’ seems to mean something different by the term. Burgess’s overarching goal

in criticizing nominalism is to show that nominalism is untenable from the naturalist

orientation—that is to say, he hopes to show that the combination of nominalism and

naturalism is untenable on his understanding of these views. Although his naturalism is

broadly Quinean in outline, he departs from Quine over the status of pure mathematics.

For Quine, pure mathematics is a form of intellectual recreation. In contrast, Burgess

(along with Maddy) regards mathematics as an important, freestanding discipline, one

that qualifies as an important component of the naturalist’s “web of belief” independently

of any applications it has to other parts of the web. Burgess’s work constitutes one

of the more serious and developed attempts to answer philosophical questions about

mathematics using what are allegedly the methodological principles of natural science and

mathematics. Ultimately I think he goes too far in this regard, for he tries to extract from

the practice of mathematics what is not there to be extracted in the first place—answers to

philosophical existence questions about mathematical objects. This is a close-cousin of the

problem I raised in the second chapter against Stewart Shapiro’s attempts to justify the

existence of ante rem structures (by his appeal to some set-theoretic existence principles

that he claims underlie mathematical practice). I believe Burgess’s criticism of nominalism

is instructive because it helps the naturalist and nominalist alike to understand to what

extent the methodological principles of natural science can help answer philosophical

questions. Moreover, it provides an interesting contrast to another well-known post-

Quinean naturalist, Penelope Maddy, whose views I examine in the next chapter.

4.2.1 Burgess’s Naturalism

Burgess’s perspective on naturalism is scattered throughout various papers spanning from

the early 1980’s to the late 2000’s. What he has to say about the naturalist position occurs

almost exclusively in papers that are primarily criticisms of nominalism. In this section I

undertake the project of reconstructing Burgess’s position on matters, using earlier and

later sources to illuminate one another.



In an early paper outlining his criticism of nominalism, he proffers the following as

evidence for platonism, and also as evidence against nominalism,

. . . science could be done without numbers. I maintain, however, that scienceat present is done with numbers, and that there is no scientific reason why infuture science should be done without them. (Burgess 2008b, 34)

Interpreted as an argument against nominalism, the idea is that the use of mathematics

in the sciences licenses accepting the existence of mathematical objects. I can imagine a

nominalist offering the following reply: “It is just question-begging to assert that science

is in fact done with numbers. I grant that science is done with mathematics, and there is

likely to never be any scientific evidence that science should be done without it, but the

conclusion does not follow unless our best understanding of the practice of mathematics

requires the postulation of mathematical objects.” However, it is hard to imagine that

Burgess would find this reply troubling; surely the practice of mathematics abounds in

existence claims, allowing Burgess to making the following riposte:

The nominalist might be able to show that mathematics could be done withoutmathematical objects. I maintain, however, that mathematics at present is donewith mathematical objects, and that there is no mathematical reason why infuture mathematics should be done without them.

So the whole problem comes back. Burgess is essentially eliciting a challenge to the

naturalist who also wants to be a nominalist: she must provide scientifically acceptable

reasons for abjuring mathematical objects, or drop her nominalist sympathies.

But what does the naturalist position come to? Writing with Gideon Rosen, naturalism

is introduced as a view that is committed

. . . at most to the comparatively modest proposition that when science speakswith a firm and unified voice, the philosopher is either obliged to accept itsconclusions or to offer what are recognizably scientific reasons for resistingthem. (Burgess and Rosen 1997, 65)

A question is immediately raised concerning the content of this characterization of natu-

ralism. What interpretation is to be given to the notion that “science speaks with a firm



and unified voice”? Surely Burgess is not suggesting that the scientific community gathers

together (or has ever gathered together) to make broad, univocally endorsed pronounce-

ments about what claims are acceptable and what claim are not acceptable. But then what

is the source of warrant in attributing to some particular claim p the universal approbation

of the scientific community? Burgess suggests that matters are to be illuminated by exam-

ining the history and practice of science. His first datum is that the evidential standards

of science are the best available and that, “there is no philosophical argument powerful

enough to override or overrule. . . scientific standards of acceptability. . . ” (Burgess and

Rosen 2005, 517). On this picture, the status of a particular claim p in relation to the scien-

tific community is to be determined using—what else?—scientific standards of warrant.

But what are the scientific standards of warrant? Burgess’s answer is that these standards

are captured by the methodological principles of the scientific community.4

Burgess is reluctant to engage in descriptive methodology of science; he views this as a

job for scientists and historians of science—not philosophers.5 With Rosen he remarks that,

“ultimately the judgment on the scientific merits of a theory must be made by the scientific

community” (Burgess and Rosen 1997, 206). Nevertheless, “what mathematical physicists

must judge in the long run, naturalized epistemologists may consider in the short run”

(Burgess 1990, 8). Burgess and Rosen enumerate a list of methodological principles guiding

theory choice very much reminiscent of Quine’s criteria of theory selection,

(i) correctness and accuracy of observable predictions

(ii) precision of those predictions and breadth of the range of phenomena forwhich such predictions are forthcoming, or more generally, of interestingquestions for which answers are forthcoming

(iii) internal rigor and consistency or coherence

(iv) minimality or economy of assumptions in various respects

(v) consistency or coherence with familiar, established theories, or wherethese must be amended, minimality of the amendment

4It should be noted that, for Burgess, there is not much difference, if any, between acceptability and belief.If scientists accept p for scientifically acceptable reasons, then naturalists are warranted in believing p.

5This disposition is the theme of two of Burgess’s papers, (Burgess 1990) and (Burgess 1998).



(vi) perspicuity of the basic notions and assumptions

(vii) fruitfulness, or capacity for being extended to answer new questions(Burgess and Rosen 1997, 209)6

Assuming that these methodological criteria capture a univocally endorsed account of

scientific warrant, it is possible to pronounce on the status of the nominalistic reinterpreta-

tions and reconstructions of mathematics. This judgment, of course, is to be made on the

basis of how well nominalist theories fare at meeting (i)-(vii). For Burgess this examination

is not exhausted by the simple-minded observation that nominalistic reinterpretations fare

well on (iv)—economy of assumptions. Rather, nominalistic reinterpretations—just like

any other theory up for adoption—must be judged on how satisfactorily they cohere with

all of these important methodological principles. And what such satisfactory coherence

consists in is in manifesting (i)-(vii) in proportion to their importance to the scientific com-

munity (as can be gleaned from the past and current decisions of scientists). Nominalists,

including modal nominalists, ought to be worried because it appears as though their views

give, “far more weight to factor (iv). . . than do working scientists” (Burgess and Rosen

1997, 210).

Should one assume with Burgess that these methodological criteria capture the uni-

versally endorsed standards of scientific warrant?7 It is far from clear. Burgess must be

granted two additional assumptions. The first assumption is that the scientific community

uniformly accepts (i)-(vii) as capturing the evidential standards of science. The second

assumption is that scientists consistently grant (i)-(vii) the same relative weight when

engaged in the process of theory selection. If the first assumption is questionable, then

it is hardly reasonable to suppose that one should ever expect to see scientists come to

universal agreement.8 To be sure, the scientific community appears to agree on the reli-

ability of low-level tools like statistical analysis and double-blind experiments. Perhaps

6Quine’s criteria are given in (Quine 1976, 247).7Thanks to Susan Vineberg for suggesting the following points.8While discussing an earlier version of this chapter, the cosmologist David Garfinkle remarked, “I’ve

never seen that happen!”



this is evidence that scientists also agree on higher-level general methodological principles.

My concern is that Burgess has not offered any evidence for the claim that scientists are

in universal agreement on the status of (i)-(vii). Similarly, Burgess offers no evidence

which would compel someone to believe that scientists are consistent in the way that they

weight (i)-(vii) when adjudicating between rival theories. But absent this evidence it is

unreasonable to be confident that the assessment of nominalist theories would betray any

inconsistency with the diverse ways in which scientists might weight (i)-(vii).

Like Burgess I am hesitant to engage in the descriptive methodology of science, and

so I do not presume to offer the final say on matters. Burgess’s account of scientific

methodology raises some unanswered questions, and some skeptics might hope to answer

them and so castigate his naturalism from the very start. One line of reasoning to which I

am sympathetic holds that science is not the coherent, unified community that Burgess

apparently takes it to be (along with most Quineans, I might add), and that the goals and

methods of science are as numerous and diverse as are the many scientific disciplines. Still,

many in the philosophical community are attracted, via Quine, to the idea that something

like (i)-(vii) describe some of the important methodological tools of working scientists.

And so, for the sake of discussion, I am willing to grant that Burgess’s naturalism is

coherent and to continue on in my investigation of whether he is justified even in believing

that his naturalism conflicts with nominalism (and whether he is justified in believing that

his platonism is consistent with his naturalism).

Returning to the dialectic, there is good reason to ask what Burgess’s naturalism has to

say about mathematical objects; science, at least on the surface, is tasked with investigating

the concrete world, and it would indeed be surprising if it ended up implying the existence

of abstract mathematical objects. Doubly surprising, too, as Burgess claims that science

is an outgrowth of common sense (Burgess 2008b, 32). Surely the claim that there exist

non-spatiotemporal objects is no teaching of common sense. Or is it? According to Burgess,

mathematics is in no way to be set apart from science; mathematics is a part of science-



proper.9 Thus mathematics and its existence claims are also outgrowths of common sense.

And since a naturalist is someone who is, “prepared to reaffirm while doing philosophy

whatever was affirmed while doing science” (Burgess 2008a, 2), naturalists are committed

to mathematical objects. On this score, Burgess says that he should be taken to

. . . affirm only (what even some self-described anti-realists concede) that theexistence of [mathematical] objects and obtaining of [mathematical] truths is animplication or presupposition of science and scientifically informed commonsense, while denying that philosophy has any access to exterior, ulterior, andsuperior sources of knowledge from which to “correct” science and scientificallyinformed common sense. (ibid.)

Nominalism, as Burgess understands the view, involves a prejudice for ontological econ-

omy that is rooted exclusively in philosophy. Thus nominalism requires “taking back”

while doing philosophy the existence claims one makes while doing science, viz., while

doing the science of mathematics. Such a prejudice is just the sort of exterior (or ulterior,

or superior) philosophizing that is disallowed under naturalism.

Before going on to discuss Burgess’s Master Argument in detail, three concerns are

worth raising. First, one could accuse Burgess of not being precise about the relationship

between the attitude scientists adopt toward a statement p and the subsequent attitude

naturalists must adopt toward p. Suppose that it is possible to make clear the conditions

under which “science speaks with a firm and unified voice.” This provides an account of

when a naturalist is compelled to accept a statement p. But nothing Burgess says informs the

discussion when attention shifts from deciding which beliefs are naturalistically required

to deciding which beliefs are naturalistically acceptable, though not required. What is a

naturalist to say about a statement p that receives from the scientific community neither

universal approbation nor universal disapprobation? Or, better yet, what is a naturalist

to say about a statement p that invites no sentiments at all from the scientific community?

Suppose a small group of scientists decry mathematics as just a useful tool, not to be taken

9Burgess and Rosen are quite clear in rejecting any view that “expels” mathematics from the “circle ofsciences” (Burgess and Rosen 1997, 211).



seriously; suppose a small group of mathematicians think that mathematics is just an

exercise in drawing derivations in various formal systems, the axioms of which need not

be taken to assert truths—what is the naturalist to say, then? Can she abjure mathematical

objects?

Second, given the obvious differences between the practice of mathematics and the

practice of empirical science, one might question Burgess’s use of the methodological

principles of empirical science in pronouncing on mathematical matters. If the methodol-

ogy of mathematics is not sufficiently similar to that of empirical science, then whether

nominalism strikes the right balance between (i)-(vii) is immaterial; what matters for

nominalist philosophies of mathematics is how well they fare when confronted with the

methods of mathematics. Of course, this is only good news for the nominalist if it turns out

that the methodology of mathematics involves sympathy for ontological economy, and on

Burgess’s view this result seems unlikely.10

Finally, one gets the feeling that somehow platonism has already been smuggled in

under the table on Burgess’s picture. If some kind of scientifically acceptable evidence is

required for eschewing commitment to mathematical objects, then by parity of reasoning

it would appear as though some kind of scientifically acceptable evidence is required

for taking on commitment to mathematical objects. Saying that such commitments are

“presupposed” in science and scientifically informed commonsense sounds like cheating.

For what is the status of this presupposition? Is it subject to acceptance or rejection on

the basis of the attitudes the scientific community adopts toward it? Burgess appears to

conflate several distinct tasks of descriptive methodology: Providing an account of the

evidential standards of science, determining what presuppositions are made by scientists,

10A brief note on the second concern: Most of Burgess’s remarks on methodology are targeted at nom-inalists who seek to replace current scientific theories with nominalized theories. Thus the charge that suchviews are to be judged by the methodological standards of empirical science is not entirely out of place.However, none of the modal nominalists that I defend have ever endorsed such a view. Thus my concernstands when one is interested in discussing the views that nominalists have actually presented. I will have anopportunity to revisit these issues in the next chapter while discussing Maddy’s views on the methodologyof mathematics.


Reflections on Burgess 193 The Arguments

and addressing what counts as “common sense” to the scientific community. Burgess

chides nominalists for doubting what is to him a basic presupposition of science—the

existence of mathematical objects. This presupposition is what appears to be doing most

of the work in Burgess’s criticisms of nominalism, and therein lies the irony. If this

presupposition is not subject to the evidential standards of science (presumably what (i)-

(vii) are intended to capture), then it hardly makes sense to complain that when nominalists

question this presupposition they are being un- or anti-scientific. On the other hand, if

this presupposition is subject to the evidential standards of science, what makes Burgess

so confident that it will pass muster? These concerns will be taken up shortly, but now I

would finally like to begin reconstructing Burgess’s arguments against nominalism.

4.2.2 The Arguments11

Burgess does not claim to provide a single argument against nominalistic reinterpretations

of mathematics. What he does instead is advance a family of arguments against the various

nominalist proposals. I begin by articulating the arguments that Burgess has actually

given so that I may show that these arguments involve a common inference. From these

arguments I then abstract a generalized argument form that I am here calling Burgess’s

“Master Argument” against nominalism. My ultimate aim is to call into question the

common inference on which all of Burgess’s arguments—including the Master Argument—

depend.

In an early paper Burgess attempts a taxonomy of nominalist positions in the phi-

losophy of mathematics (2008b). Burgess allows the nominalist three possible positions.

First, the nominalist could have her sights on replacing existing scientific theories with new

11A note on terminology: (Paseau 2005) correctly points out that the terms ‘reinterpretation’ and ‘re-construction’ refer to different activities. Reinterpretation is, “interpretation different from the standardinterpretation, which is the interpretation of mathematics accepted by mathematicians when doing math-ematics” (ibid., 378). Reconstructions, “change the actual mathematics as opposed to its interpretation”(ibid.). Nevertheless, Burgess and Rosen, along with most commentators, use the terms ‘reinterpretation’ and‘reconstruction’ interchangeably as blanket terms encompassing both reinterpretation and reconstruction inPaseau’s senses. Below I shall attempt to be clear about which criticism attaches to which sort of activity; thereader should keep in mind that doing this, in certain circumstances, involves (charitably) distorting theactual arguments Burgess has given.



nominalistic theories that do not quantify over mathematical objects. Such a nominalist is

called a revolutionary reconstructionist. Second, the nominalist could have her sights on

uncovering the ultimate meaning of mathematical assertions. A nominalist who alleges

that, when properly understood, mathematical assertions do not quantify over mathe-

matical objects, is called a hermeneutic reinterpretationist. Finally, the nominalist could be

advancing the view that mathematical and scientific theories are just useful fictions. Such

a nominalist is called an instrumentalist.12

4.2.2.1 Against Hermeneutic Reinterpretation

The thesis of hermeneutic reinterpretation is that,

1. Nominalistic interpretations of mathematics unveil the true meaning or “deep struc-

ture” of mathematical assertions.

But for Burgess, “as a thesis about the language of science, hermeneutic nominalism

is. . . subject to evaluation by the science of language, linguistics” (ibid., 36). And although

he is no linguist, he is happy to announce that such a view is, “seldom entertained by

students of language,” and assures the reader that the hermeneutic thesis lacks the kind of

evidence that would convince linguists of its truth (ibid). Thus it is clear Burgess believes

that,

2. There is no relevant (linguistic) scientific evidence supporting hermeneutic nominal-

ism.

In defending her views the hermeneutic reinterpretationist is thereby committed to de-

fending an “implausible hypothesis in linguistics” (ibid., 41). It should be plain from the12Burgess is rather curt in dismissing instrumentalism, claiming that, “[t]he philosopher who begins by

rejecting theoretical physics as fiction will find no logical place to stop, and in the end will be unable, withoutfurther inconsistency and self-contradiction, to accept commonsense belief as fact” (ibid., 32). As none ofthe major nominalistic projects advance the sort of instrumentalism that cascades down this slippery slope,I will pass over this argument just as curtly as it was introduced. Burgess’s criticism of instrumentalismshould be read more as a critique of Bas van Fraassen’s views on the philosophy of science than as anobjection to any particular nominalist philosophy of mathematics. Some of Burgess’s mature views onmathematical instrumentalism will be touched upon in the closing discussion of Burgess’s platonism. Toanticipate as-yet undefined terminology, Burgess now recognizes a hermeneutic version of instrumentalismknown as “attitude-hermeneuticism.” More on this below.



previous discussion of Burgess’s naturalism that this means that,

3. Hermeneutic nominalism is thereby un- or anti-scientific,

which clearly entails that,

4. Hermeneutic nominalism is unacceptable to someone who adopts the philosophical

orientation of naturalism.

The following conclusion is unavoidable,

5. Hermeneutic nominalism and naturalism are inconsistent doctrines.

And that is why, as a naturalist, Burgess rejects hermeneutic nominalism.13

4.2.2.2 Against Revolutionary Reconstruction

The thesis of revolutionary reconstruction is that,

6. Nominalized scientific theories ought to be adopted in favor of their platonistic

counterparts.

As I have already remarked, Burgess (with Rosen) worries that defending this thesis

places too much importance on the value of ontological economy (Burgess and Rosen

1997, 210). For instance, there is good reason for supposing that a paper on nominalistic

physics would never be accepted by a physics journal (Burgess 2008b, 37).14 Moreover, the

adoption of nominalized theories would likely hinder the progress of science. There would

be great costs associated with revising scientific curricula. New breakthroughs would be

less frequent due to the adoption of more complicated theoretical apparati (Burgess 2008b,

38-9). Thus Burgess seems confident that,

13Burgess no longer believes that the argument adduced in this section presents the most powerful caseagainst hermeneutic nominalism. His current view is that hermeneutic reinterpretations are to be understoodas paraphrases of ordinary mathematical language and that paraphrase is not an acceptable vehicle foreschewing commitment to mathematical objects. Since I have discussed these matters already in the secondchapter of this dissertation, I will not pause to rehearse and respond to Burgess’s ulterior case againsthermeneutic reinterpretation.

14This point is reiterated with Rosen in (Burgess and Rosen 1997, 210).



7. There is no relevant scientific evidence supporting revolutionary nominalism.

This means that,

8. Revolutionary nominalism is thereby un- or anti-scientific.

And the rest of the argument proceeds as in the case of hermeneutic nominalism:

9. Revolutionary nominalism is unacceptable to someone who adopts the philosophical

orientation of naturalism.

10. Revolutionary nominalism and naturalism are inconsistent doctrines.

And that is why, as a naturalist, Burgess rejects revolutionary nominalism.15

4.2.2.3 Against Tract Housing

In his most recent collaboration with Rosen, the catch-all categories of “hermeneutic

nominalism” and “revolutionary nominalism” are expanded and refined (Burgess and

Rosen 2005, 517). The hermeneutic position is split up into attitude-hermeneutic nominalism

and content-hermeneutic nominalism. The revolutionary position is split up into naturalized

revolutionary nominalism and alienated revolutionary nominalism.16

15Would Quine follow Burgess here? Arguably not. If revolutionary nominalism turns out to be the “bestoverall theory” then Quine is obliged to profess nominalism. That adopting revolutionary nominalismmight hinder the progress of science speaks against the plausibility of regarding revolutionary nominalismas candidate for the “best overall theory.” Whether this is convincing evidence against the possibility ofrevolutionary nominalism turning out to be the “best overall theory” depends on whether Quine wouldfollow Burgess in holding that theories must be judged according to whether they manifest criteria (i)-(vii) inproportion to how scientists wield (i)-(vii). Of course, Quine ultimately came to reject nominalism becausehe believed that it was not possible in principle to eschew commitment to mathematical objects. Quine didnot reject nominalism because he believed nominalist physics would be theoretically uneconomical; Quinenever required that the practicing scientist actually use the best overall theory while doing science. This,then, in another way in which Burgess departs from Quine.

16This refined terminology comes from Rosen, as Burgess acknowledges in his introduction to (Burgess2008a, 4). Nevertheless he appears to endorse the terminology and cites Rosen as an influential figure (ibid.,3). He also cites Rosen as “correctly” pointing out that instrumentalism, “itself comes in a revolutionaryversion (this is the attitude philosophers ought to adopt) and a hermeneutic version (this is the attitudecommonsense and scientific thinkers already do adopt)” (ibid., 4). I will pass over these subtleties as they in noway affect what I have to say in the remaining sections of this chapter. For the record, Burgess indicates thathe now associates the hermeneutic position of (Burgess 2008b) with Rosen’s content-hermeneutic nominalism,and he also says that hermeneutic instrumentalism should be associated with attitude-hermeneutic nominalism(Burgess 2008a, 4). For a defense of revolutionary instrumentalism, see (Daly 2006).



The content-hermeneutic position is identical to the position known simply as “hermeneu-

tic nominalism” above; a content-hermeneutic nominalist believes that, deep down, math-

ematical assertions are not really about mathematical objects. Similarly, the naturalized-

revolutionary position is identical to the position known simply as “revolutionary nominal-

ism” above; a naturalized-revolutionary nominalist believes that ontological economy is a

guiding methodological principle of science. Since I have already given Burgess’s reasons

for rejecting these views I will move on and briefly discuss the other two positions. (It is

worth noting that some nominalists have described themselves as “revolutionary.” For

instance, Mary Leng advocates a view she sometimes calls “revolutionary fictionalism.”

But as discussed below, she does not attach the same sense to the term as does Burgess;

her revolution is a revolution in the understanding of mathematics and not in its practice.

Thus, Burgess’s arguments against “revolutionary nominalism” should not be taken to

apply to all nominalists that call themselves ‘revolutionaries.’)

Alienated-revolutionary nominalists seek to replace scientific standards of warrant with

philosophical standards of warrant. In this way they advocate the abandonment of

the naturalistic orientation and argue that the justificatory standards of mathematics,

though acceptable to scientists, are not really acceptable after all. Such a view is clearly

inconsistent with Burgess’s naturalism. As my interest here is in examining Burgess’s

objections to nominalistic views that purport to cohere with the philosophical orientation

of naturalism, I will give just the bare outline of what Burgess has to say against the

alienated-revolutionary. The alienated-revolutionary nominalist is typically motivated by

the apparent epistemological quandary identified in (Benacerraf 1973). Since mathematical

objects are acausal, it is unclear how it is possible for human beings to have knowledge

about them. Burgess thinks that this concern depends critically on causal theories of

knowledge and that there are good reasons for rejecting such theories. Moreover, any

puzzle about how mathematicians come to possess justified mathematical beliefs is to be

solved by examining the actual means by which mathematicians arrive at their beliefs. Just



as physicists can avail themselves of the prima facie reliability of perception (and so can

avoid having to refute the external-world skeptic), mathematicians may avail themselves

of the prima facie reliability of mathematical beliefs (and so can avoid having to refute the

mathematical-knowledge skeptic). On Burgess’s view, the best hope for the alienated-

revolutionary is to achieve a stalemate concerning the ultimate justificatory status of

mathematics.17 It is unclear what sort of evidence would be acceptable to both parties, one

way or the other, on the topic of whether mathematics needs a philosophical foundation.

The attitude-hermeneutic nominalist argues that the fact that scientists and mathemati-

cians accept mathematical assertions should not be taken to imply that scientists and

mathematicians believe in the face-value truth of these assertions. For instance, a set

theorist while on the job may sincerely utter a statement implying the existence of an

inaccessible cardinal, but the attitude-hermeneutic nominalist will maintain that, upon

reflection, this set theorist may consistently maintain that there are no such things as

sets. As before, the question of what to make of this position comes down to whether it

is recommended by the available evidence. Prima facie, the evidence speaks against the

attitude-hermeneuticist:

. . . when a person understands a sentence S, confidently affirms it withoutqualification and without conscious insincerity, organizes serious activity justas would be done if it were believed, and so on, then we have a powerful casefor attributing to that person a belief that S. . . (Burgess and Rosen 2005, 526)

Absent a direct disavowal of existence assertions on the part of scientists (including

mathematicians—but apparently, not including formalists or Alfred Tarski), the conclusion

is drawn that there is not sufficient evidence for recommending the adoption of the

attitude-hermeneutic position. It should be clear how to construct an argument against the

attitude-hermeneutic position similar to those given in the previous two sections.

17See (Burgess and Rosen 1997, 26-49) and (Burgess and Rosen 2005, 523).



4.2.2.4 The Master Argument

On the assumption that the above revolutionary and hermeneutic strategies exhaust the

viable options for the nominalist, Burgess concludes that nominalism is untenable. It

should be obvious, however, that the arguments against both the hermeneutic nominalist

and the revolutionary nominalist take the same form. Here are the first two arguments

again:

Against (content) hermeneutic nominal-

ism:

Against (naturalized) revolutionary nomi-

nalism:

1. Nominalistic interpretations of mathe-

matics unveil the true meaning or “deep

structure” of mathematical assertions.

6. Nominalized scientific theories ought

to be adopted in favor of their platonistic

counterparts.

2. There is no relevant (linguistic) sci-

entific evidence supporting hermeneutic

nominalism.

7. There is no relevant scientific evidence

supporting revolutionary nominalism.

3. Hermeneutic nominalism is thereby un-

or anti-scientific.

8. Revolutionary nominalism is thereby

un- or anti-scientific.

4. Hermeneutic nominalism is thus un-

acceptable to someone who adopts the

philosophical orientation of naturalism.

9. Revolutionary nominalism is thus un-

acceptable to someone who adopts the

philosophical orientation of naturalism.

5. Therefore hermeneutic nominalism and

naturalism are inconsistent doctrines.

10. Therefore revolutionary nominalism

and naturalism are inconsistent doctrines.

There is much these arguments have in common. Each share the same basic structure, and

it is this structure that I am here calling Burgess’s “Master Argument” against nominalism.

The Master Argument looks like this:


Reflections on Burgess 200 Replies in the Literature

11. Nominalists advance thesis T .

12. There is no scientific evidence in support of T .

13. It follows that T is un- or anti-scientific.

14. Thus T cannot be accepted by a naturalist.

15. Therefore nominalism and naturalism are inconsistent doctrines.18

In a short while I will I make a case for thinking that Burgess’s naturalism is incapable

of validating two of the Master Argument’s important inferences—the inferences from (12)

to (13) and from (13) to (14). In order to motivate this response I first examine several of

the ways in which nominalists have responded to Burgess’s arguments. I then indicate

why I think that most of these responses are unsatisfactory.

4.3 Replies in the Literature

Of the three modal nominalists examined in the first chapter, two—Charles Chihara

and Geoffrey Hellman—have offered responses to Burgess’s criticisms. These responses

come in two varieties. First are the replies, stemming primarily from Chihara, to the

various claims Burgess has made about scientific methodology and the scientific merits of

nominalistic reinterpretations. Second are the replies that Burgess’s subdivisions (including

Rosen’s refinements) mischaracterize the rationale behind nominalist philosophies of

mathematics and therefore fail to pronounce on any of the views that philosophers like

Chihara and Hellman have actually proposed.19 Both kinds of responses purport to unveil

Burgess’s arguments as unsound.

18Cf. Burgess and Rosen’s “schematic argument” for accepting the existence of mathematical objects (2005,516-7).

19Hartry Field should be included in this list, too. Though often one of Burgess’s targets, I am not awareof any place in Field’s work where has bothered to respond to Burgess’s arguments. For this reason I giveField’s fictionalism very little attention in this chapter.


Reflections on Burgess 201 Scientific Merits

4.3.1 The Scientific Merits of Nominalistic Reinterpretation

A common point of emphasis in Burgess’s writings is that nominalistic reinterpretations

and reconstructions of mathematics would not be regarded as progress by scientists

(Burgess and Rosen 1997, 210). Thus it would be sophomoric to expect to see a work

on nominalism published in the Physical Review. Burgess’s suggestion seems to be that

the cash value of nominalistic reconstruction and reinterpretation is to be determined by

practicing scientists.

Chihara notes that most of the nominalistic literature is self-consciously aimed at

responding to the Quine/Putnam indispensability arguments. Seen in this context, Chihara

asks,

Are empirical scientists or mathematicians best equipped intellectually andby training to deal with the many and complex philosophical issues that sucha dispute may involve? Should they be entrusted with the final word on themerits (scientific, logical, and philosophical) of such responses? Would sendingpapers detailing these reconstructions to physics journals constitute “true tests”of the merits of these works? (2004, 163)

He responds that an editor’s decision to reject a paper on nominalism would not so

much betray ontological economy as a second-class methodological principle as it would

reflect the fact that, “the journal’s readership would not be an appropriate audience for a

philosophical paper of that sort” (ibid.).20 Chihara’s accusation is that Burgess mislocates

the significance of nominalistic reinterpretation in viewing it as an attempt at science

rather than as an attempt at philosophy. Nevertheless Burgess regards the underlying

assumption of the indispensability arguments—that the dispensability in principle (as

opposed to the dispensability in practice) of mathematical entities suffices to demonstrate

their non-existence—as a concession to nominalism that he does not feel compelled to

20Alexander Paseau claims that, “the question of an adequate construal of mathematics is not consideredby science journals” (2007, 139). If Paseau is right, this is not just bad news for nominalism, it is bad news formathematics as a whole! I suppose Burgess’s case could be improved by shifting the thought experimentfrom science journals to mathematics journals and pointing to cases of published articles that discuss mattersof interpretation. Paseau is skeptical of Burgess’s prospects.



make (Burgess and Rosen 1997, 213). For, granting this concession would imply that

ontological economy is an overriding principle of scientific methodology, a move that

Burgess clearly desires to avoid.

Perhaps in anticipation of this response, Chihara has made efforts to show how his

Constructibility Theory can produce advances in science and mathematics. He begins with

an analogy involving nonstandard analysis, an alternative form of analysis that introduces

hyperreal numbers.21 Although the nonstandard model of the continuum is thought by

some mathematicians to be, “so counter-intuitive as to be repugnant,” it turns out that

in nonstandard analysis, “the basic concepts of the calculus can be given simpler and

more intuitive definitions” (Chihara 2007, 62).22 Doctoral research carried out by Kathleen

Sullivan found that,

[T]here does seem to be considerable evidence to support the thesis that thisis indeed a viable alternative approach to teaching calculus. Any fears on thepart of a would-be experimenter that students who learn calculus by way ofinfinitesimals will achieve less mastery of basic skills have surely been allayed.And it even appears highly probable that using the infinitesimal approach willmake the calculus course a lot more fun both for the teachers and for students.23

21A number is a hyperreal number if it is either an infinite number or an infinitesimal. An infinite number isone that is greater than any sum of finite numbers. A number is an infinitesimal when it is the multiplicativeinverse of an infinite number.

22Chihara gives the example of continuity. Consider the following definition of continuity from EdwardGaughan’s Introduction to Analysis (Pacific Grove, CA: Brooks/Cole Publishing Company, 1998), 83:

Suppose E ⊂ R and f : E 7→ R. If x0 ∈ E, then f is continuous at x0 iff for each ε > 0, there is aδ > 0 such that if

|x− x0| < δ, x ∈ E,

then

|f(x)− f(xo)| < ε.

Compare this to Robinson’s nonstandard definition given in (Chihara 2007, 63):

A function f is said to be continuous at c iff:

(i) f is defined at c; and

(ii) whenever x is infinitely close to c, f(x) is infinitely close to f(c).

I shall leave it to the reader to decide which of these definitions is simpler and more intuitive.23“The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach,” American Mathemat-

ical Monthly 83 (1976), 375. As quoted in (Chihara 2007, 64).



If Sullivan’s findings are to be trusted, nonstandard analysis has a pedagogical value over

and above any interest mathematicians might have in the subject.

Chihara hypothesizes that Burgess would see things differently if he were to approach

the case of nonstandard analysis in the same way that he approaches nominalism. What is

nonstandard analysis good for?

Evidently, there are two principal answers that should be considered:

(A) Nonstandard analysis provides us with a hermeneutical ‘analysis of theordinary meaning of scientific language’. . .

(B) It provides us with an alternative version of science, which is better than,and to be preferred to, our present-day versions of science. . . (Chihara2007, 64)

Neither (A) nor (B) constitute attractive options. There is inadequate linguistic evidence

for the hermeneutic option (A). And the revolutionary option (B) would involve a costly

revision of the mathematical curriculum. He then gives the following Burgess-inspired

conclusion:

Since anti-nonstandard analysts reject all the hermeneutic and revolutionaryclaims of the nonstandard analysts, from their viewpoint, nonstandard analysis isdistinct from and inferior to standard analysis. What is accomplished by producinga series of such distinct and inferior theories? No advancement of science proper,certainly. . . (ibid., 66, Chihara’s emphasis)

Chihara intimates that this is a ludicrous way of looking at nonstandard analysis because

doing so fails to capture the pedagogical value it possesses, to say nothing about the actual

uses of nonstandard analysis in applications.24 His suggestion is that assessing the value

of nominalistic reconstructions in this way is similarly ludicrous.

Before responding to this analogy I would like first to point out that Burgess would

very likely agree with Chihara on the various merits of nonstandard analysis. Indeed,

papers on nonstandard analysis are routinely published in mathematics journals—what

24For a brief list see (ibid., 67).



better evidence could Burgess ask for? I make this point purely as a clarification; I do not

imagine Chihara was attempting to divine Burgess’s views on nonstandard analysis.

If this analogy is supposed to show that it is possible for Chihara’s Constructibility The-

ory to contribute to one’s understanding of mathematics and science, then it is important

for him to show that the case of nonstandard analysis is actually analogous to the case of

Constructibility Theory. But I cannot help thinking of all the important—that is, impor-

tant from Burgess’s perspective—disanalogies between the case of nonstandard analysis

and Chihara’s Constructibility Theory. To begin with, the basic concepts of nonstandard

analysis can be clearly and succinctly articulated.25 The basic concepts of Constructibility

Theory include inadequately studied modal notions.26 Second, nonstandard analysis was

formulated by a mathematician—Abraham Robinson, whereas Constructibility Theory

was created by a philosopher. Although of freestanding significance only to those adopting

a crass scientism, this nevertheless is prima facie evidence that the theories were constructed

with entirely different motives, serve entirely different goals, and invoke entirely different

methodological principles and standards of evidence. Or, at the very least, it is evidence

that nonstandard analysis and Constructibility Theory stress different aspects of the goals

of the scientific community. In either case the burden is on Chihara to demonstrate that

his project is one which complements the scientific enterprise, rather than one which runs

contrary to it. In line with this point is the observation available to Burgess that nonstan-

dard analysis contributes to the education of mathematicians and serves important roles

in scientific applications. One would therefore expect to see Chihara point to cases where

Constructibility Theory has contributed to the education of mathematicians and to cases

where the theory has played an important role in scientific applications of mathematics.

Unfortunately I can find no example of Constructibility Theory being used in these ways,

25History aside. Infinitesimals were quite controversial postulations during the development of thefundamental ideas of calculus.

26This may not be a relevant point of disanalogy if the conclusion is supposed to be that ConstructibilityTheory is unscientific. To show this one need only point to the sundry of poorly articulated concepts inparticle physics, quantum mechanics, and many other scientific disciplines.



aside from Chihara’s hopeful (though yet unconfirmed!) speculation that some day it may

help in

. . . getting one to have doubts about a dominant way of understanding some-thing [that] can free the mind to explore other alternatives—even lead to thediscovery of a new paradigm—opening the way to any number of importantinsights (just as erasing an entry in a partially completed crossword puzzle canopen up a flood of new possibilities). (ibid., 69)

At best, Chihara has shown that his view has the ability to address many of the philosophical

problems one encounters in the philosophy of mathematics—problems that Burgess, for

better or worse, is not much troubled by. To be sure, one of these philosophical problems

is the “problem of application,” that is, the problem of how to explain why mathematical

reasoning is so successful in scientific applications. Chihara goes to great lengths to

show how his view can recover such applications (2004, ch. 9). But on my understanding

of Burgess, seeking after the recovery of mathematical applications is a fool’s errand.

Mathematics is applied successfully in science, end of story. For Burgess the question is

not, “Can this view of mathematics recover applications of mathematics?,” but instead,

“Can this view of mathematics be applied profitably in science?” That Chihara can show how

Constructibility Theory handles applications is evidence that the theory can, in principle,

actually be put to use in applications, but it is difficult to imagine that it would provide a

practical or economical framework in which to carry out scientific reasoning.

If the remarks of this section are interpreted as a case that Constructibility Theory can,

by Burgess’s standards, contribute to the advancement of science, my judgment is that

Chihara is unsuccessful. However, these remarks might suggest an alternative conclusion,

one which holds that Burgess misplaces the significance of what it means to say that a

philosophical position succeeds only in advancing the philosophical debate. I will take up

this clue after discussing two other responses to the Master Argument.


Reflections on Burgess 206 Scientific Merits Redux

4.3.2 Scientific Merits Redux: Chihara and the Attitude-Hermeneuticist

The attitude-hermeneuticist contends that scientists and mathematicians can be consis-

tently described as not fully believing in mathematical existence assertions despite the

fact that, according to Burgess, the same scientists and mathematicians find the same

assertions acceptable by mathematico-scientific standards. Burgess replies that scientists

and mathematicians accept existence assertions without any conscious reservations and

that the default presumption in this situation is that scientists and mathematicians believe

these existence assertions—a presumption that nominalists have not defeated.

Chihara does not deny that mathematicians both accept and believe assertions such as

16. The null set exists.

Instead he argues that Burgess’s criticism of the attitude-hermeneuticist fails to account for

the context in which mathematicians accept existence assertions like (16). Such assertions

imply the existence of mathematical objects only if they are, “true statement[s] about what

exists in the actual world” (Chihara 2006, 323). But assertions like (16) are never accepted

in a vacuum; (16) is a theorem of Zermelo Frænkel set theory. Thus,

Just because a mathematician assents verbally to (16) ‘without conscious silentreservations’, one cannot conclude that she believes (16) to be actually true. Herassent could consist merely in the acceptance that (16) is true in a relative sense,e.g., is truly a theorem of ZF or is true for the iterative concept. So much morewould have to be argued to yield the conclusion that almost all mathematiciansand scientists accept (16) as actually true. (ibid., 325)27

Notice that this response does not rest on any novel hermeneutical claim about assertions

like (16); that (16) is a theorem of ZF is an uncontroversial truth Chihara takes at face-value.

Resolving whether mathematicians accept the axioms of ZF as truths of the world—as

opposed to, say, assumptions that yield fruitful research—is a further matter.28 Burgess27I have made inessential changes in numbering.28Cf. David Corfield’s observation that, “were we to ask a mathematician today whether he considered a

system of axioms to be true, he would most probably tell us that he thought it had interesting models or thatit described an important construction” (2003, 182). Also, cf. Maddy’s set-theoretic naturalism examined inthe next chapter.


Reflections on Burgess 207 Scientific Merits Redux

floats the reply that the axioms of well-studied mathematical theories enjoy an immunity

from skepticism that is analogous to the justificatory status of perceptual judgments;

“perceptual judgments are justified by default, innocent until proven guilty,” and so

mathematical theories are justified by default, innocent until proven guilty (Burgess and

Rosen 2005, 523). If this remark is on point, it serves only as an attempt to shift the burden

of proof and fails to provide any explanation for why mathematical theories occupy a

privileged justificatory status and why this status conveys mathematical theories as true

simpliciter.

Chihara’s intent throughout is to undermine the claim that there is no scientific evidence

on record for thinking that scientists and mathematicians adopt an attitude other than

belief in existence assertions as truths about the world. What evidence does he offer for

thinking that scientists may adopt an attitude other than belief on matters?

I have been struck by the fact that, in discussing the nominalism-Platonism dis-pute with non-philosophers, the scientists among the discussants are frequentlysurprised, if not downright amazed, to learn that there are many eminentcontemporary mathematicians and logicians who maintain not only that theaxioms of mathematical theories such as set theory and number theory are gen-uine assertions about the real world, but also that there exist in the real worldsuch things as sets, functions, numbers, spaces, and the like. The impression Ihave gotten, as a result of such discussions, is that it had never occurred to thesescientists that mathematical axioms—especially the existential ones—can be reasonablyunderstood to have such metaphysical implications. (Chihara 2004, 289, emphasisadded)

If these anecdotal remarks are indicative of the general attitude that the scientific com-

munity espouses toward mathematics, then there is in fact good evidence that scientists

believe that mathematical theories are not ultimately truths about the real world, vitiat-

ing Burgess’s claim that the existence of mathematical objects is a “presupposition” of

science.29 This unveils the second premise of the attitude-hermeneutic instance of the

29David Liggins reaches a related conclusion in (Liggins 2007). Liggins understands Burgess and Rosen asaiming to show that, “philosophers are required to believe those existence theorems which are acceptable bymathematical and scientific standards” (ibid., 107). He responds that, “if there actually have been competentmathematicians who did not accept the existence theorems, this tends to undermine the idea that accepting


Reflections on Burgess 208 The False Dilemma Reply

Master Argument (that there is no scientific evidence in favor of adopting the thesis of

attitude-hermeneuticism) as false. As well, it speaks against the idea that economy of

mathematical ontology is entirely foreign to the scientific community as a methodological

principle.

I am inclined to agree with Chihara on these matters; he appears to have shown that

Burgess’s argument against the attitude-hermeneuticist includes a false premise (line (11)

from the Master Argument). Moreover, the above remarks suggest a further point. If it

is questionable that scientists believe in mathematical theories as truths about the real

world, then the platonist must elicit some kind of evidence against the contrasting view

that scientists only believe in existence assertions as theorems of mathematical theories.

The trouble is that producing this kind of evidence is likely to involve the advancement

of positive theses about the meaning of mathematical assertions, and in doing this one

is liable to utilize overtly philosophical views about language and ontology that are not

evidently principles of scientific methodology. Might Burgess’s platonism be inconsistent

with his naturalism? I take up this issue in the closing section of this chapter, after I give my

reasons for thinking that the Master Argument is not validated by Burgess’s naturalism.

4.3.3 The False Dilemma Reply

The success of Burgess’s Master Argument against nominalism depends on the plausibility

of generating a comprehensive list of nominalistic theses T1, . . . , Tn, showing that any

given nominalist is committed to at least one of these, and arguing that for each such Ti

there exists a corresponding instance of the Master Argument. A popular reply is that

neither the hermeneutic nor revolutionary positions described above provide accurate

characterizations of the nominalist philosophies of mathematics. That is, Burgess argues

from a false dilemma when he requires that nominalists colonize his subdivisions.

Both Chihara and Hellman have made the False Dilemma reply. In particular, they

them is required by the standards of mathematics” (ibid., 110). Formalist mathematicians are given as asupporting example.



have both addressed Burgess’s characterization of their views as being either hermeneutic

or revolutionary. In his response, Chihara claims to, “accept neither of the alternatives

he allows his opponents” (1990, 189). Meanwhile, Hellman alleges that, “this dichotomy

leaves no room for what I should have thought was the proper category for most of these

programs, namely a kind of ‘rational reconstruction’ ” (1998, 342).

Chihara explains in broad terms the motivation behind his Constructibility Theory:

The nominalistic reconstructivists of the sorts I have in mind do not attemptto judge common sense and science from some higher, better, and furtherstandpoint. They seek to piece together their account of mathematics in a waythat is compatible with both what science teaches us about how we humansobtain knowledge and also what we already know about how humans learnand develop mathematical theories. Furthermore, these nominalists do notreject mathematics—a fortiori, they do not reject mathematics on the basis of“some higher and better and further standpoint”. On the contrary, their goal isto understand the nature of mathematics in a way that is compatible with theother features of the Big Picture they are attempting to construct. (2004, 159)

Several pages later, he identifies in specific terms what he is up to:

. . . I was proposing an answer to a highly theoretical and deeply philosophicalquestion: can our contemporary scientific theories be reformulated or recon-structed in a way that will not require the assertion or the presupposition ofabstract mathematical objects? This is a modal question. It is not a practicalquestion of how best to teach physics in our secondary schools, colleges, anduniversities. (ibid., 165-6)

Clearly, Chihara is not a revolutionary nominalist; his concern is to rebut the indispens-

ability arguments and not at all to offer a substantive proposal for how mathematics (and

science) ought to be done. Chihara also explains that he is not a hermeneutic nominalist,

stating that his theory, “was not meant to be an analysis of the mathematical statements

asserted by practicing mathematicians” (ibid., 165).30 What is the proper setting in which

to approach Constructibility Theory? According to Chihara, his, “work is directed at

philosophers who put forward ontological claims based upon the use of mathematics in30One might worry that if Chihara is not interested in analyzing the meaning of mathematical assertions

then he will not be able to “understand the nature of mathematics.” I raise a related worry at the end of thischapter.



empirical science” (1998, 319). Burgess, of course, would not be satisfied by this reply. As

has been seen, for Burgess the true authority of mathematical existence claims comes not

from the empirical sciences but from the mathematical sciences.31

The purpose of Hellman’s Modal Structuralism, “is to help answer certain metamathemat-

ical or metascientific questions, not normally entertained in pure and applied mathematical

work proper” (Hellman 1998, 342). He elaborates,

If a reconstruction is not intended as a replacement or even revision of ongoingtheory, if it is intended rather as a coexisting proposal for understanding suchtheory or its accomplishments, preserving its substance while facilitating ac-commodation within a naturalistic epistemology, then it would indeed be fairlypreposterous for the author of such a reconstruction to submit it to a physics ormathematics journal. (ibid., 344)

And elsewhere he insists that,

. . . nominalistic reconstructions need be neither hermeneutical nor revolutionarybut can be—and in most cases in question are—preservationist while attemptingto solve or avoid certain epistemological, metamathematical, or metascientificproblems not (or not yet) treated within science itself. (Hellman 2001a, 703)

Thus, Hellman does not see his view (or other nominalist views) as fitting neatly into

Burgess’s subdivisions.

These observations are spot-on. Neither of the major nominalist projects in the philoso-

phy of mathematics can be accurately classified either as hermeneutic or as revolutionary;

the disjunction Burgess poses to the nominalist is indeed a false dilemma.32 Thus, this

response soundly refutes Burgess’s original arguments. However, I am not convinced

that this response is enough. Why? Because it seems perfectly reasonable to suppose

that Burgess can always enlarge the scope of his list of nominalistic theses, making the

whole problem come back. Whatever the best characterizations are of the motives and

31Cf. Chihara’s defense of attitude hermeneuticism in the previous section.32A similar defense is available to Field, who is also self-consciously motivated by the indispensability

arguments. Field does not describe his fictionalism as a view about how mathematics or science ought to bedone, but instead as a demonstration of the possibility of doing mathematics and science without quantifyingover mathematical objects. One might, however, accuse Field of hermeneuticism in describing mathematicalknowledge as logical knowledge.


Reflections on Burgess 211 My Reply

goals behind nominalistic reinterpretations—and I worry with Burgess about how clearly

nominalists have articulated themselves on these details—Burgess will always have re-

course to the reply that the very fact that scientists are not engaging in such activity is

all the evidence he needs in order to demonstrate that nominalist philosophies of mathe-

matics are unscientific. In Chihara’s case, Burgess can quite reasonably voice skepticism

about whether there exist any scientific reasons which might motivate expending one’s

intellectual resources examining the modal question about whether it is possible to avoid

commitment to mathematical objects. That is, unless Chihara can convince scientists of the

scientific importance of asking his modal question, it appears as though Burgess can easily

formulate a new instance of his Master Argument against Chihara-style reconstructions.

Similar remarks are in order against Hellman. That he seeks answers to questions “not

normally entertained” in mathematics is kindling for the fire; Burgess will see this as just

more evidence that Hellman’s goals are unscientific.33

Simply put, there is something else wrong about Burgess’s reasoning that has yet

to be addressed in the literature.34 Each of the above replies maintains, with varying

degrees of success, that the published instances of Burgess’s Master Argument contain

false assumptions. I know of no serious attempts to show that Burgess’s naturalism is

incapable of validating some of the key inferences of the Master Argument—a possibility I

would like to raise in the next section.

4.4 My Reply

I argued in the previous section that a demonstration of the scientific merits of nominal-

istic reconstruction requires, on Burgess’s terms, more than showing that it is possible to

carry out scientific reasoning in nominalistic theories; nominalists must point to actual

cases of scientists putting their theories to use. I also argued that advancing the False

33For skepticism about whether a project like Hellman’s even qualifies as metascientific, see (Marquis2000).

34As I show below, there is, nevertheless, an appropriate setting in which to advance the False Dilemmareply.


Reflections on Burgess 212 My Reply

Dilemma reply requires more than showing that Burgess has mischaracterized the aims

of reinterpretation. His criticism is charitably interpreted as beginning from a justifiable

position of unclarity concerning what nominalist accounts of mathematics have to say

about the actual content of mathematics. Burgess’s subdivisions constitute his efforts

toward understanding what nominalists are up to when they present novel constructions

and interpretations of mathematics. And so, first, nominalists must precisely locate the

arena of discourse in which nominalist reinterpretations and reconstructions of mathe-

matics are put forward. And second, such a discussion must occur within an organic

extension of scientific methodology. It is prima facie unlikely that nominalists will succeed

in these endeavors, and so some suitable version of the Master Argument is bound to run

its course. How, then, is a nominalist to respond to Burgess’s arguments, if she wishes to

do so without rejecting his naturalism in toto?

Here again is the argument; perhaps on closer inspection a previously overlooked error

will be uncovered.

11. Nominalists advance thesis T .

12. There is no scientific evidence in support of T .

13. It follows that T is un- or anti-scientific.

14. Thus T cannot be accepted by a naturalist.

15. Therefore nominalism and naturalism are inconsistent doctrines.

I suggested above that there is likely to be some possible instance of (11) that a nominalist

would accept, and that given such a starting point, it is unlikely that she can show that

(12) is false. And that is all Burgess needs in order to get an instance of the argument

up and running. But attacking (11) or (12) is the only way to show that the argument

contains a false premise. The only possible response left to the nominalist is to argue

that the argument is invalid. But how to proceed? It is clear that the inference from (14)


Reflections on Burgess 213 The Tonsorial Question

to (15) is beyond dispute. This leaves the inferences from (12) to (13) and from (13) to

(14). These two inferences are the heart and soul of Burgess’s criticism of nominalism.

If either should fail, the Master Argument is invalid. I have intimated at various points

above that my intention is to show that Burgess’s position on the relationship between the

attitude scientists adopt toward a statement p is, at best, equivocally related to the attitude

naturalists must adopt toward p. More specifically, the equivocality of this relationship

has as a consequence that Burgess’s naturalism is singularly incapable of validating at the

same time both the inference from (12) to (13) and the inference from (13) to (14). In making

good on this promise it should be instructive to examine Burgess’s clearest account of the

relationship between the attitudes scientists adopt toward a statement p and the attitudes

naturalists must adopt toward p, which is presented in (Burgess 1998).

4.4.1 The Tonsorial Question

For Burgess, the success of nominalist reinterpretations and reconstructions of mathematics

turns on the status of Ockham’s razor. If ontological parsimony is a governing principle

of scientific methodology, then the nominalist can, in principle, use Ockham’s razor to

validate her various Ti’s as scientific; otherwise she faces the chopping block. Burgess

maintains that, “nominalism is no teaching of science” (2005, 90). His account of why this

is so should make it as clear as is possible precisely how he uses the terms ‘scientific’ and

‘unscientific’.

In Burgess’s writings one finds an apparent inconsistency. On the one hand, he holds

that Ockham’s razor is an acceptable scientific principle, agreeing that, “explanations in

terms of extraordinary agencies are not to be resorted to until explanations in terms of

ordinary agencies have been exhausted” (Burgess 1998, 210). This is explained as the idea

that scientists avoid gratuitous assumptions, be they ontological or ideological (ibid., 211).

On the other hand, Burgess also clearly maintains that science is chin-deep in abstract

objects, writing (with Rosen) that, “a thoroughgoing naturalist would take the fact that

abstracta are customary and convenient for the mathematical (as well as other) sciences to



be sufficient to warrant acquiescing in their existence” (Burgess and Rosen 1997, 212). But

if scientists accept Ockham’s razor, and the nominalistic reinterpretations of mathematics

succeed in eschewing commitment to mathematical objects, why is Burgess not compelled

to conclude that the “customariness” and “convenience” of mathematical objects in science

make only an illusory gesture at their existence?

The key to dissolving this apparent inconsistency is to recognize, with Burgess, that

Ockham’s razor is a general principle that can be applied in any number of distinct ways.

Ockham’s razor can be wielded to evict the mathematical realm of anything except sets, or

except functions. It can be wielded to banish unnecessary causal agencies. It can be wielded

to avoid nonsense postulations, such as the invisible purple dinosaur that undetectably

manipulates the firing of neurons in a way that convinces certain philosophers of the

existence of mathematical objects. (Why is the imagined beast always purple?) The

nominalist condones the use of Ockham’s razor as a general rule calling her to avoid

unnecessarily postulating extraordinary agencies, to be sure, but she is also interested in

putting it to use voiding the blank ontological checks that scientists and mathematicians

write themselves while doing science and mathematics.35 According to Burgess, whether

scientists care about the economy of mathematical ontology, “must be tested directly against

the evidence of past decisions of physicists” (1990, 11).

How, then, does Ockham’s razor fare as a scientific principle tasked with dispensing

with mathematical objects? Burgess is skeptical of this all the way back in 1983 when “Why

I am not a Nominalist” was first published:

. . . the avoidance of ontological commitments to abstract entities does not seemto have won recognition in the scientific community as being in itself a goal ofthe scientific enterprise on a par with scope and accuracy, and convenience andefficiency, in the prediction and control of experience. It seems distinctivelyand exclusively a preoccupation of philosophers of a certain type. (Burgess

35Recently, at least one nominalist has explicitly endorsed Ockham’s razor as a scientific principle thatfavors nominalism. Mary Leng writes that, “adopting our ordinary scientific standards of inquiry surelyrequires us to adopt the principle of Ockham’s razor. . . I feel justified. . . in moving beyond mere agnosticismand concluding that we are justified in denying the existence of mathematical objects” (2010, 260)—but cf.(Balaguer 1998, 144-8).



2008b, 37)

Seven years later, he again voices his reservations about the scientific value of the nominal-

ist’s use of Ockham’s razor:

. . . since rigor and consistency are already usually conceded by descriptivemethodologists to be weighty scientific standards, the burden of proof seems tobe more on those who would insist that ontological economy of mathematicalapparatus is also a weighty scientific standard. This burden of proof has notyet been fully met. (Burgess 1990, 12)

Nevertheless it is not until 8 years later that Burgess finally provides evidence for these

claims. Quoting Gideon Rosen, he asserts that, “purging physics of a commitment to

numbers serves no goal recognized either implicitly or explicitly in the practice of science”

(Burgess 1998, 204). The lesson of (Burgess 1998) is that the philosopher’s use of Ockham’s

razor is importantly distinct from the scientist’s use of that same principle. Scientists,

Burgess claims, implement Ockham’s razor to rid the world of unnecessary causal agencies.

The nominalist, meanwhile, uses Ockham’s razor to rid the world of unnecessary abstract

entities. With Rosen, Burgess asks, “has abstract ontology ever been what was at issue in

any important case of dispute between proponents of rival theories in empirical science?”

(Burgess and Rosen 1997, 217). Their answer? That they, “know of no clear example of

striving after economy of abstract ontology in any domain of science, and we are dubious

that there is one” (ibid., 225). The outlook is even worse when turning to the mathematical

sciences:

It is not part of the practice of ordinary mathematics to take steps to disavowbelief in the standard existence theorems, to warn students against believing,or the like. Thus we lack what would be the best kind of direct evidence thatthe practice of mathematics and science involves something less than belief inexistence theorems. (Burgess and Rosen 2005, 526)

That students of mathematics are not warned “against believing” is rather poor evidence

for Burgess’s conclusion; the resources that are useful for getting students to understand

and prove theorems are not necessarily those resources which best highlight the nature



of mathematics. Nevertheless, absent the direct evidence in question, “it strains credulity

to suggest that the internal norms governing scientific inquiry demand such disavowal

nonetheless” (ibid., 528). The main point underlying these remarks is best summarized in

(Burgess and Rosen 1997, 213):

. . . reconstructive nominalists who profess ‘naturalism’ have not themselvesmuch said it. They have not proceeded by first presenting studies of thedistinctions and divisions observed within the community of working scientists,and then citing these as warrant for discarding pure mathematics and ignoringfamiliarity, perspicuity, and fruitfulness. So one may well ask what the sourceof their warrant is supposed to be.

In essence, Burgess’s complaint is that there is simply no evidence that scientists, includ-

ing mathematicians, condone the use of Ockham’s razor for eschewing commitment to

mathematical objects.36 The nominalist is alleged to commit the fallacy of accident when

she presumes it is right and good to use this otherwise acceptable principle of scientific

methodology in felling numbers and sets (of course, the reader has no assurances that

Burgess has himself conducted a scientific study of science in making this allegation).

In order to avoid falling into this trap, nominalists must first uncover evidence that

avoiding commitment to abstract objects is a governing principle of scientific methodology.

Burgess does not expect that the nominalist will be successful in this endeavor. Moreover,

even if the nominalist were to be successful in convincing the likes of Burgess that eschewing

commitment to mathematical objects is a going scientific concern, she faces the subsequent

task of uncovering evidence that avoiding commitment to abstract objects is an important

enough criterion of theory-selection to compete with other well-established criteria such as

familiarity and scope. It is at this point that Burgess can entertain the question as to whether

nominalist reinterpretations of mathematics might ever win the approval of the scientific

community; most nominalists are even willing to grant that the answer to this question

36Cf. Stewart Shapiro: “The serious point underlying Burgess and Rosen’s suggestion is that only scientists(including editors of professional scientific journals) are to determine what counts as scientific merit. Andfor a naturalist, what else counts as merit? The fact is that scientists are not much interested in eliminatingreference to mathematical objects. The nominalist has to show them that, by standards which they haveimplicitly adopted, scientists should eschew reference to mathematical objects” (2000, 247-8).



is ‘no.’ The upshot is that nominalist philosophies of mathematics and the concerns that

motivate them are thoroughly unscientific. (If Chihara is to be trusted, many scientists and

mathematicians espouse agnosticism toward the ontological commitments of mathematics.

Agnosticism is a form of disavowal, and so there is at least some evidence suggesting that

scientists and mathematicians adopt an attitude that amounts to less than full belief toward

existence theorems. Burgess could suggest that these scientists and mathematicians are

guilty of not faithfully applying their own standards of evidence. However, Burgess is

already on record as stating that the existence of mathematical objects is a presupposition

of science. Is this presupposition subject to confirmation or disconfirmation on the basis

of evidence? If not, then agnosticism toward it is not to be denigrated on the basis of the

evidential standards of science. On the other hand, if this presupposition is subject to

confirmation or disconfirmation on the basis of evidence, then the presence of agnosticism

in the scientific community ought to count as evidence against it. This, I take it, is Chihara’s

response on behalf of the attitude-hermeneuticist. These concerns will be revisited in the

last section of the chapter.)

But in this context, what does the term ‘unscientific’ really mean? And what is Burgess

implying when calling a claim ‘unscientific’? Burgess has made a strong case that the

particular fashion in which nominalists wield Ockham’s razor does not emblematize any

going principle of scientific methodology, apparently confirming the assumption of the

Master Argument that there is no scientific evidence in support of the theses advanced by

nominalists. He subsequently advertises nominalism as un- or anti-scientific. This appears

to recommend the following meaning of the expression ‘p is un- or anti-scientific’:

17. A statement p is un- or anti-scientific if and only if p has not been justified by any

evident principles of scientific methodology.

But this characterization is much too strong; it rules as un- or anti-scientific all unrecognized

consequences of current accepted scientific theories, to say nothing of what it implies about

the status of statements belonging to any provisional theories or to any superior theories



that may be developed in the future. And it is thoroughly unclear how plausibly this

principle applies to episodes of theory change. Moreover, Burgess does not pretend to

have performed an exhaustive study of the history of scientific methodology; perhaps,

unknown to philosophers, there are cases of scientists using Ockham’s razor to pare down

abstract ontology. A moderate revision to (17) fixes this error:

18. A statement p is un- or anti-scientific if and only if p can be justified only by principles

that are not evidently principles of scientific methodology.

This definition does a better job of encapsulating Burgess’s misgivings about nominalism.

He indeed asserts that no nominalist has actually produced a scientifically acceptable

justification for reconstructing mathematics, moreover he is convinced that all possible

justifications for the available nominalist reconstructions are likely to cite views about

ontology or ontological commitment that are entirely foreign to the scientific community.

(Presuming that (a) good sense can be made of who in particular belongs to the “scientific

community,” and (b) nominalist philosophers of mathematics are (for the most part) not

members of the “scientific community.” Ostensibly Burgess would consider someone to

be a member of the scientific community when they carry out research in some scientific

discipline while adopting the methodological principles of natural science, apportioning

the same weight to criteria (i)-(vii) as do other members of the scientific community

(which poses a potential circularity issue). Nominalist philosophers of mathematics are

allegedly disqualified because they place too much weight on the criterion of economy

of abstract ontology.) Now, (18) does not imply that as-yet underived consequences of

current accepted theories are un- or anti-scientific; nor does it imply that statements of

provisional theories or as-yet undeveloped superior theories are un- or anti-scientific.37

Most importantly, however, is that (18) is sufficient to justify the inference from (12) to (13)

in the Master Argument. Can it also secure the inference from (13) to (14)?

37This latter point may only be true provided that the class of evident principles of scientific methodologyis permitted to vary as science progresses.



Recall again Burgess’s understanding of naturalism as a view that commits one

. . . at most to the comparatively modest proposition that when science speakswith a firm and unified voice, the philosopher is either obliged to accept itsconclusions or to offer what are recognizably scientific reasons for resistingthem. (Burgess and Rosen 1997, 65)

This appears to recommend the following position on when a statement is naturalistically

acceptable:

19. If p is univocally endorsed by the scientific community, then acceptance of p is

naturalistically required.

As things stand, (19) only implies that p is naturalistically unacceptable when not-p is

univocally endorsed by the scientific community. Hence the following position on when a

statement is naturalistically unacceptable:

20. If p is univocally rejected by the scientific community, then rejection of p is naturalisti-

cally required.

To some, (19) and (20) might appear too strong; nothing essential would be lost by moving

their antecedents into the subjunctive mood:

21. If p is such that it would be univocally endorsed by the scientific community, then

acceptance of p is naturalistically required.

22. If p is such that it would be univocally rejected by the scientific community, then

rejection of p is naturalistically required.38

In order to get from (13) to (14), Burgess must show that nominalism is unscientific in a

sense that satisfies the antecedent of either (20) or (22). It turns out that it is to Burgess’s

benefit to focus just on (22); no comprehensive poll of the scientific community has ever

taken place on the realism debate, so it would seem that there is not sufficient evidence

38Perhaps both of these subjunctive conditionals should end with: “. . . of an ideal observer.”



for holding that nominalism is actually the object of the universal disapprobation of the

scientific community—(20) is at best vacuously true.39

According to (18), to say that p is unscientific means that p cannot be justified by any

principle of scientific methodology. This could mean one of two things. In the first place,

it could mean that p implies the contradiction of some statement q univocally accepted

by the scientific community (for instance, an explanation given by an astrologer for why

a relationship ended poorly). Call this the strong reading of (18). Whenever a statement

p is unscientific according to the strong reading of (18), it is clear that p ought to be

univocally rejected by the scientific community, and hence supports the inference from

(13) to (14). In the second place, it could mean that p, though unsupported directly by

scientific methodology, is nevertheless justified in such a way as not to interfere with any

of the pronouncements of science (for instance, the utilitarian association of happiness

with pleasure). Call this the weak reading of (18). Whenever a statement p is unscientific

according to the weak reading of (18), there can be no prima facie claim made about how p

is viewed by the scientific community. But then (22) (even if it is restated as a biconditional)

does not imply that such a statement is naturalistically unacceptable.40 And so claims that

are unscientific only according to the weak reading of (18) fail to validate the inference

39Perhaps there is reason for thinking that (22) is also vacuously true. Earlier, while motivating Burgess’snaturalism, I briefly recounted the criticism that it is a distortion of the practice of science to suppose that thescientific community would ever come to uniform agreement about nontrivial matters. If this criticism canbe sustained, then (22) will be non-vacuous in very few cases. But then, for most statements p, rejection of p isnot naturalistically required. On this picture, sympathy to nominalism is naturalistically permissible becausesympathy to nearly everything is naturalistically permissible! Below I try to flesh out a more attractiveaccount of why sympathy for nominalism is consistent with Burgess’s naturalism.

40Not even (19) and (21), reformulated as biconditionals, imply that claims satisfying the weak sense of(18) are naturalistically unacceptable; all that is implied is that naturalists are not compelled to accept suchclaims. In a paper arguing that trump naturalism (accept p if science sanctions p) does not imply biconditionalnaturalism (accept p if and only if science sanctions p)—principles analogous to (19) and (21) and theirbiconditional reformulations—Paseau comes to a very similar conclusion: “Silence regarding p—neithersanctioning p nor sanctioning ¬p—is not the same as sanctioning suspension of belief about p” (2010, 647).Paseau’s discussion is not sensitive to the distinct ways in which a claim could fail to gain the acceptanceof the scientific community. He offers theological and astrological claims as examples of claims that, “maybe acceptable even if science does not sanction [them]” (ibid., 643). Nevertheless I suspect that astrologyand theology are inconsistent with science at the level of methodology, in the sense that science rules againstthe evidential standards of astrology and theology, whereas it is by no means clear that a similar verdict iswarranted in the case of nominalist philosophies of mathematics.



from (13) to (14).41

An important worry remains: Why suppose that anything satisfies the strong reading

of (18)? Or, what comes to the same question, why suppose that anything fails to satisfy

the weak reading of (18)? What solace is to be had in discovering that nominalism satisfies

only the weak reading of (18) if it turns out that just about everything else satisfies this

reading as well? The nominalist appears to need, along with Burgess, some mechanism by

which she can ensure that the scientific community comes to uniform agreement about at

least some things. Without such a mechanism it is empty to use terms like “naturalistically

acceptable” and “naturalistically required.” This is no small task—indeed, it might be

thought of as the greatest obstacle facing anyone interested in developing a coherent

methodological naturalism. My aspirations are not so ambitious. Although the scientific

community is not necessarily unified on general matters like methodology, I am going to go

out on a limb and assume that the scientific community is unified on many matters of fact

about the physical world: that there exist electrons, that the actual physical constants are

approximately what science says they are, that planetary conjunctions are not importantly

causally related to interpersonal relationships, etc.

I submit, accordingly, that a preliminary test of whether a methodological principle

is naturalistically unacceptable is whether its implementation produces in its advocates

beliefs that conflict with scientifically accepted facts. I claim that this unveils most theolog-

ical and astrological belief-forming mechanisms as unacceptable. Nominalist philosophies

of mathematics involve no such tomfoolery. Of course, this is not a sufficient defense of

nominalism. Naturalism is not only a view about what one should believe but also a view

about how one should form beliefs. Consider the epistemological freak Trusci, who for no

discernible reason is able to reliably form beliefs about the microphysical world that are

later confirmed by particle physicists. Trusci acquires no belief that conflicts with any mat-

41Cf. Balaguer’s defense of fictionalism in (Balaguer 2009). Balaguer argues that the thesis of fictionalismis mathematically unimportant and that philosophers—not mathematicians—are best equipped to assess itsmerits.



ter of fact, but intuitively, a naturalist should not adopt belief in p solely on the authority of

Trusci. The case of Trusci proves that although one can and should reject unreliable methods,

one should not accept just any reliable methods. I am not certain how to rule out wonders

like Trusci. But how Trusci forms beliefs is altogether distinct from the possible routes

nominalists, including modal nominalists, envision for the epistemology of mathematics.

Nominalists are motivated by attempting to account for mathematical knowledge using

already familiar and naturalistically acceptable faculties, such as the human competence

for recognizing patterns and for making logical and modal inferences. And so it strikes me

as unreasonable in the highest degree to group nominalist philosophies of mathematics

together with the likes of Trusci. Reasons for thinking that Trusci’s belief-forming methods

satisfy the strong reading of (18) do not transfer to the philosophical considerations which

underlie nominalist philosophies of mathematics. Nonetheless, I suspect that there do

exist reasons for thinking that Trusci’s belief forming methods satisfy the strong reading

of (18), and so I do suppose that the scientific community would uniformly disapprove

of Trusci-followers—thus I believe that the scientific community does come to uniform

agreement on some things that are more robust than mere matters of fact. I admit that this

belief is one that I am not certain how to justify in a rigorous fashion.

So what has Burgess shown? Has he unveiled nominalistic reinterpretations as satisfy-

ing the strong reading of (18)? Or has he only shown that nominalistic reinterpretations

satisfy to the weak reading of (18)? Revolutionary strands of nominalism clearly satisfy

the strong reading of (18), but no nominalist that I know of has ever endorsed a revolu-

tionary view. The case against the hermeneuticist is less clear. Would mathematicians and

scientists univocally assent to the platonist’s semantical analysis of mathematical existence

assertions? If so, then content-hermeneuticists satisfy the strong reading of (18). Would

mathematicians and scientists univocally assent to the view that mathematical existence

assertions are truths about the world (as opposed to truths as theorems of mathematical

theories)? If so, then attitude-hermeneuticism satisfies the strong reading of (18). But, as



the saying goes, these are big ‘ifs’; mathematicians and scientists are known to adopt a

variety of views about the nature of mathematics, and it would be naıve to expect universal

agreement on these matters. Nevertheless, it is not clear that any nominalist that I defend

in this dissertation is engaging in either kind of hermeneutical project. Here, then, is the

appropriate venue for advancing the False Dilemma reply. Burgess has not accurately de-

picted any of the nominalist views he claims to reject. A fortiori, he cannot, with confidence,

assert that these views are unscientific according to the strong reading of (18). This last

point gains traction in conjunction with the observation that nominalistic reinterpretations

of mathematics are designed not to interfere with the day-to-day practices of scientists and

mathematicians. At best, Burgess has a case that nominalism satisfies the weak reading of

(18); but then he has failed to show that nominalism is un- or anti-scientific in any sense

that compels a naturalist to reject nominalism!

David Liggins reaches an assessment of Burgess’s arguments that appears to be very

similar to my own (Liggins 2007).42 He gives the following gloss on Burgess’s account of

naturalistic acceptability:

23. When mathematical and scientific standards require mathematicians and scientists

to accept an existence theorem t, philosophers are justified in accepting t regardless

of what philosophical arguments are offered against it. (Liggins 2007, 107)

Liggins remarks the last clause of (23) assumes a deference to science that many philoso-

phers will find “too much to take,” and that a more plausible principle should read,

24. When mathematical and scientific standards require mathematicians and scientists

to accept an existence theorem t, philosophers are justified in accepting t unless there

is a sufficiently strong philosophical argument against it. (ibid., 110)

42Liggins’s explicit target is Burgess and Rosen’s positive argument for accepting the existence of mathe-matical objects, as opposed to their negative arguments against nominalism (Burgess and Rosen 2005, 516-17).Ostensibly these arguments depend on the same views about when a naturalist is compelled to accept aclaim.



Against both (23) and (24) he argues that the acceptance of existence theorems is not

required by the standards of mathematics. He maintains that, e.g., formalist mathemati-

cians may reject existence theorems without thereby abandoning or countermanding the

evidential standards of mathematics. Of course, most mathematicians do not identify

themselves as formalists; many in fact accept existence theorems without reservations of

any kind. Although the statements that mathematicians are required to accept may fail to

speak against nominalism, perhaps some of the statements mathematicians are permitted to

accept discredit nominalism. To deal with this wrinkle, Liggins entertains one final gloss

on naturalistic acceptability:

25. When mathematical and scientific standards permit mathematicians and scientists to

accept an existence theorem t, philosophers are justified in accepting t unless there is

a sufficiently strong philosophical argument against it. (ibid., 111)

But (25) holds philosophers to higher standards than those to which mathematicians

and scientists are held; it implies that statements optionally accepted by scientists are

nevertheless required for philosophers.43 Liggins alleges that (25) goes, “beyond mere

respect for the internal standards of mathematics,” and is not consonant with Burgess’s

“comparatively modest” naturalism (ibid).

How does Liggins’s response compare to mine? His main contention is that the stan-

dards of mathematics do not require mathematicians to accept the platonist’s interpretation

of mathematical assertions; his example of formalist mathematicians is offered as evidence

that there is actual mathematical evidence in favor of anti-platonist interpretations of

mathematics. In order for this to be convincing evidence that naturalist philosophers are

permitted to adopt formalist views on mathematics, something like the following principle

must be assumed:

26. When mathematical and scientific standards permit mathematicians and scientists to

43And worse; if there is some proposition such that acceptance of both p and ¬p is permitted in science,then philosophers are obliged to accept both p and ¬p!



accept t, philosophers are permitted to accept t as well.

I do not find anything particularly objectionable about (26); I welcome evidence suggesting

that acceptance of nominalism is mathematically and scientifically permissible.44 Never-

theless my reply is distinct in relying on a principle that is even more generous than (26). I

have argued that even if there is no evidence that scientists and mathematicians would

accept a claim p, it does not thereby follow that p is naturalistically unacceptable, provided

that p does not imply the contradiction of anything scientists and mathematicians would

uniformly accept (i.e., provided that p satisfies (18) on only the weak reading). I submit

that Liggins’s response, rather than competing with mine, serves instead to complement it.

However, my response has the advantage of not depending on any (actual or hypothetical)

cases of mathematicians and scientists accepting nominalism.

Mary Leng’s response to Burgess is also quite similar to my response. Leng aims to show

that the revolutionary fictionalist, although advancing a revolution in the understanding of

the practice of science, nevertheless proposes no revolution in the practice of science. After

attempting to undermine Burgess’s positive arguments for the existence of mathematical

objects, she writes that the fictionalist’s

. . . scepticism about literalism as the default interpretation of mathematicsweakens Burgess’s position here, since it is not entirely clear that the revolu-tionary fictionalist is denying anything that mathematicians and scientists havesincerely asserted. (Leng 2005, 282)

If she is right, Leng has shown that revolutionary fictionalism satisfies only the weak

reading of (18). What conclusion does she allege follows from this observation?

It is, we can accept, a mistake to advocate the abandonment of a successful disci-pline on the basis of philosophical scruples about, for example, the ontologicalassumptions of that discipline. To this extent, we may be modest. However,

44But here I must say that I would like to hear quite a bit more about the conditions under which scientificand mathematical standards permit—without requiring—practitioners to adopt beliefs. Without a preciseaccount of such conditions one cannot be confident that, e.g., the formalist beliefs of certain mathematiciansare actually permissible as far as scientific and mathematical standards are concerned. Cf. my discussion inchapter five of how Maddy’s uses mathematical methodology as a rein on metaphysical theorizing aboutmathematics.



so long as we can explain why a practice that is in some respects misguided isnevertheless successful, we need not advocate the abandonment of that practicejust because it falls short of some of our expectations. (ibid., 283)

On the assumption that the revolutionary fictionalist is indeed successful in vindicating the

utility of mathematics, both pure and applied, Leng asserts that fictionalism is compatible

with accepting the practices of mathematics and science. This reply is offered only on

behalf of the revolutionary fictionalist—very well; then I have indicated the way in which

a similar response can be made on behalf of many other kinds of nominalist projects.

I have argued that Burgess has not done enough to show that nominalism is unscientific

in a damaging way. I have done so by putting pressure on an aspect of his view about which

he is unclear: the naturalistic acceptability of statements that receive from the scientific

community neither universal approval nor universal disapproval. The onus is on Burgess

to articulate a more refined notion of naturalistic acceptability that provides an intuitively

palatable account of why nominalism should be thought of as inconsistent with naturalism

despite the purgatorial status of nominalism in relationship to the scientific community.

Surely there exist many naturalistically acceptable (and unacceptable) claims that receive

from the scientific community neither universal approval nor universal disapproval, and

there ought to be a way of sorting the ripe from the rotten. Burgess’s naturalism is

insufficiently equipped to handle this task, and so is not equipped to betray nominalism

as naturalistically unacceptable. Paseau picks up on this theme, writing that,

The debate must be decided not by table-thumping declarations from eitherside that the question is or is not scientific, but by examining the exact boundaryscience itself posits between questions within the scientific realm and thoseoutside it. (2007, 144)

Saying these things does not, however, suffice to show that naturalism and nominalism

are indeed consistent. The nominalist still must clearly articulate the motives and goals of

her reinterpretations, moreover, she would do well to seek out evidence that hers is the

kind of view that at worst satisfies only the weak reading of (18), and at best plays some


Reflections on Burgess 227 Why Burgess is not a Platonist

positive role in contemporary science and mathematics.

4.5 The One Hope for the Vanquished: Why Burgess is not a (Moder-

ate) Platonist

Sinon’s false promise of peace has been heeded, despite Laocoon’s fear of Greeks (even

those bearing gifts!). The great false horse has been drawn within the city gates. Achaeans

soon disembark from the bowels of the wooden beast and terrorize the city. Troy is

engulfed in flames. Aeneas utters the following words to his few remaining warriors: una

salus victis nullam sperare salutem.45

Suppose, for the sake of argument, that the remarks of the previous section are wildly

implausible, and that Burgess’s criticism of nominalism does not rest on an insufficiently

articulated naturalist position. That is, suppose that the Master Argument is permitted

to run its course against the nominalist, much as the Trojans foolishly took Sinon at his

word. Then the nominalist has no choice but to admit defeat; her views are unscientific in

such a way that she cannot maintain her nominalism and still call herself a naturalist. But

a glimmer of hope remains; though her views are unscientific, perhaps her adversary fairs

no better. Perhaps there is hope for neither the nominalist nor the platonist.

Burgess describes himself as a moderate platonist. Moderate platonism is that the view

that when scientists and mathematicians accept a claim that appears to posit the existence

of mathematical objects, e.g., “there exist infinitely many prime numbers,” that one ought

to thereby acquiesce to the existence of mathematical objects. Moderate platonism is to

be preferred to nominalism because nominalism is unscientific. However, this preference

for moderate platonism requires for its justification some kind of evidence that the view

is not unscientific in any important sense of the term. Unfortunately, by Burgess’s own

lights, moderate platonism counts as unscientific. In reply to the hermeneutic nominalist,

he alleges that,

45The one hope for the conquered is hoping for no hope.



. . . there is a considerable problem of evidence for any claim about “depth”analysis. And perhaps one needs to be especially careful about evidence whenthe claim is made not by a linguist for whom an understanding of hiddenmechanisms of language is an end in itself, but rather by a philosopher withulterior ontological motives. (Burgess 2001, 440)

Burgess should find these remarks troubling; platonism—even moderate platonism—is just

as entangled in claims about “depth analysis” as is hermeneutic nominalism, and Burgess

never once points to linguistic evidence in favor of his interpretation of mathematical

language. Chihara presses this point nicely:

. . . where is the evidence that supposedly supports belief in the entities? Here,I mean evidence of the sort that would convince the physicist or biologist.Certainly, Burgess supplies none at all. So it is hard to see why he is confidentthat these mathematical objects exist. And if he is able to provide some kind ofphilosophical reason for holding on to such a belief, despite the absence of anyscientific evidence, it is hard to see why he should claim that his position is thescientific one and that his opponent’s position is superstitious and dogmatic.(1990, 189)

What is more, an important part of Burgess’s evidence that nominalism is unscientific

consists in expressing his doubts that nominalistic interpretations of mathematics and

physics would ever be published in physics and linguistics journals. The conclusion

drawn is that eschewing commitment to mathematical objects is not a going concern of

science. But Burgess should be the first to admit that in order for platonism to count as

scientific, he must evince evidence that the postulation of mathematical objects is a going

concern of science. Chihara’s insight here is incisive: “If, as Mathematical Realists claim,

mathematical objects must be postulated to account for physics, it is striking that physicists

do not publish papers in which the existence of mathematical objects is postulated” (1998,

319). Leng adds, “we might label Burgess’s alternative view ‘hermeneutic literalism’, and

wonder whether this view fares any better than hermeneutic fictionalism as an account of

what mathematicians really mean” (2005, 280).

There is hope yet for Burgess, and that is to show that the attitudes scientists and

mathematicians adopt toward mathematical ontology will somehow favor his platonism.



If scientists see their work as confirming the existence of mathematical objects, then the

preference for moderate platonism counts as scientific, other things being equal. Unfortu-

nately, Burgess explicitly discounts the testimony of scientists as reliable evidence on these

matters:

. . . there may be serious difficulties with the methodology of pestering scientistsfor opinions on philosophical issues to which they may have given little or nothought, and accepting their answers as indicative of their intentions in puttingforward the affirmations that they do put forward when philosophers leave thescene and let them get back to work. (Burgess 2008c, 55)

This is curious. Burgess asks the nominalist to believe that the decisions of the editors

and referees of scientific journals are capable of apportioning the value of economy of

mathematical ontology, while also insinuating that these editors and referees are not

especially well-qualified to address the question of ontology.46 Things only get worse

when it is remembered that Burgess holds that striving after economy of mathematical

ontology is, “a matter to which most working scientists attach no importance whatsoever”

(Burgess 2008b, 37). Penelope Maddy agrees with this last point, stating that, “scientists feel

free to adopt any mathematical apparatus that is convenient and effective, without concern

for its abstract ontology” (2005b, 451).47 What is difficult to comprehend is why anyone

should be compelled to believe that this general lack of concern for abstract ontology

is unequivocally evidence for platonism. Why, for instance, should the carefree attitude

scientists take toward abstract postulations not count as evidence for nominalism? It is

not unreasonable to believe that the very fact that scientists are so carefree about their use

of mathematics is evidence that they do not have the same attitude about mathematical

existence theorems as they do about the existence of, e.g., subatomic particles.48 Absent

a good reason for discounting the carefree ontological attitudes of scientists as evidence

46Even those who are qualified, “tend to disagree with each other quite as much as professional philoso-phers do” (ibid.).

47Cf. (Chihara 2004, 289).48John Halpin suggested (in conversation) that the carefree attitudes taken by scientists toward abstract

objects actually constitute evidence for nominalism.



for nominalism, no conclusion should be drawn about whether these attitudes support a

particular philosophical position on the existence of mathematical objects. In the present

context (in which the validity of the Master Argument is granted), the upshot is that

platonism and nominalism are equally unscientific, and hence both views are inconsistent

with Burgess’s naturalism.

If the opinions of scientists are inconclusive (or worse, if they end up showing that any

ontological position is unscientific), what other considerations could move a naturalist

to adopt the platonist’s position? In response to the rise in popularity of fictionalist

accounts of mathematics, Burgess remarks that what is, “in actual fact very doubtful

is whether mathematicians who assert that there are prime numbers greater than 101010

intend their assertion only as something ‘non-literal’ ” (2008c, 54). Burgess’s case here is

quite clear. Mathematicians assert that ‘there exist numbers greater than 101010 ,’ and there

is no evidence that they mean anything other than that there exist numbers greater than

101010 . But for Burgess the literal assertion that ‘there exist numbers greater than 101010’ is

not to be understood by appealing to Fregean considerations about truth and singular

terms. This would be unscientific, not to mention question-begging. Rather, Burgess

understands the term ‘literal’ to be indicative of what one is not doing:

The force of “literally” is not to assert that one is doing something more besides,but to deny that one is doing something else instead: meaning something otherthan what one says, as when one speaks metaphorically. . . One does not haveto think anything extra in order to speak literally; one has to think somethingextra in order to speak non-literally. (ibid., n. 9)

What is more, there is, in general, a preference for understanding assertions literally:

. . . the “literal” interpretation is not just one interpretation among others. It isthe default interpretation. There is a presumption that people mean and believewhat they say. It is, to be sure, a defeasible presumption, but some evidence isneeded to defeat it. The burden of proof is on those who would suggest thatpeople intend what they say only as a good yarn, to produce some actualevidence that this is indeed their intention. (ibid., 54)

This is a restating of the idea that the existence of mathematical objects is a “presumption”



of science. In reply to this, it is worth remarking for a second time that mathematical

existence assertions are never uttered in a vacuum. Existence assertions are licensed by

theorems, the ultimate warrant of which come from the axioms of mathematical theories;

it is certainly no distortion of mathematical practice to understand existence assertions

in this context. Focusing on lone existence assertions, then, is a poor test of whether the

evidential standards of science support this presumption; it would be more appropriate to

investigate whether mathematicians ever seriously endorse the literal truth of the axioms

of mathematical theories.

Nevertheless, a question remains about just what the literal (or “standard”) interpre-

tation of a mathematical assertion comes to, and why it should be supposed that the

existence of mathematical objects is presupposed under such an interpretation. David Cor-

field observes that in the mathematical development of the groupoid concept (important in

category theory and homotopy theory), “arguments for the ‘existence’ of groupoids did not

figure in the array surveyed. . . mathematicians make no use of the idea in their advocacy

of the groupoid concept” (2003, 230). If this observation generalizes, then considerations

relating to the existence of mathematical objects need not be seen as underlying a literalist

treatment of mathematical language. Nevertheless, Hellman is skeptical about whether

any interpretation deserves the title “standard interpretation”:

Of course, natural science takes mathematics for granted and uses it oppor-tunistically without questioning its foundations or its interpretation. But thissuggests to me that it does not worry about how to interpret mathematics at all,not that it accepts in any considered way a face-value, literal, platonist readingof mathematics. Even working classical mathematicians don’t universally agreeon such a reading. Most scientists, I would wager, have never even thoughtabout the issue. We should not, therefore, even say that “platonistic mathemat-ics” is practically indispensable; at best we may be able to say that the compact,standard languages of mathematics are practically indispensable. This doesnot tell us that any particular reading or interpretation of such languages isrequired. (2001a, 703)

Alexander Paseau voices a related skepticism,



Which norms, after all, have been successful when it comes to questions of inter-pretation? Patently, mathematical norms have been effective in the generationof successful mathematics; but the question is whether they have been effectivein the generation of successful interpretations of mathematics. That they havea better track record here than philosophical norms remains to be seen. (2005,392)

. . . the standard interpretation of mathematics does not command the sameallegiance as the cherished tenets of our worldview. Only the most deludedphilosopher will deny the Moorean observation that we are more certain that2+3=5 (and other mathematical truths) than any philosophy that tells us other-wise. But certainty about the mathematical content of the statement that 2+3=5does not extend to certainty about any metaphysical claim the statement mightmake. It would not extend, for instance, to the claim that there really existnon-spatiotemporal, acausal objects standing in the addition relation. (ibid.)

It is unclear what Paseau intends to communicate by using the term ‘mathematical content.’

What exactly is the mathematical content of the statement ‘2+3=5’ and how can this content

be captured without giving an interpretation to the symbols ‘2’, ‘3’, ‘5’, ‘+’, and ‘=’?

He responds to the objection that the standard content of such assertions is platonistic

(Paseau 2007, 146-9). Nevertheless he provides very little information about what that

content is (other than that it is somehow captured equally well by nominalist and platonist

interpretations). This is another instance of a worry I have voiced at several points

above concerning the ability (or willingness) on the part of nominalists to provide an

account of what mathematics is about. (Not that I am suggesting they are alone in this

struggle; if it is true that the platonistic interpretation of mathematical language is optional,

then presumably such an interpretation does not explicate the content of mathematical

assertions in greater detail than nominalistic interpretations.) It is one thing to show that

the practice of mathematics is indifferent to the interpretation of mathematical language,

but until a view becomes available that delineates mathematical content in precise terms,

an important piece of the puzzle is missing.

Nevertheless there is a wide body of evidence suggesting that mathematics, contra

Burgess, does not come with its own interpretation. And so without something like



Frege’s analysis of mathematical language, it is not clear precisely how one proceeds from

the endorsement of a mathematical assertion to the existence of mathematical objects.

The question is this: Without the additional assumption of some kind of platonism,

why is the presumption in favor of the literal a presumption that has anything at all

to do with ontology? When push comes to shove, Burgess always falls back on this

presumption. However, he is happy to discount the philosophical musings of scientists

and mathematicians concerning the ontological presuppositions of their theories. But what

other kind of evidence is admissible for a naturalist? All that is left over are philosophical

principles and philosophical intuitions. And that is my point—one does not come to accept

the existence of mathematical objects unless one views science and mathematics through

platonistic goggles. But then even moderate platonism needs for its support theses that are

not emblematic of any principles of scientific methodology: the same deadly accusation

alleged to extirpate nominalism. And that is why Burgess is not a (moderate) platonist.


234

Chapter 5

Reflections on Maddy

5.1 Introduction

The upshot of my examination of Burgess is that his particular interpretation of naturalism

does not, in the end, have the resources to speak against nominalist theories of mathemat-

ics, including the modal nominalist theories I defend in this dissertation. As a defense of

modal nominalism, my response to Burgess relied on the on the assumption that modal

nominalism is not unscientific in the sense that it requires the rejection of any claims uni-

vocally accepted by the scientific community. Relativized to mathematics, the assumption

is that modal nominalism is not unmathematical in the sense that it requires the rejection

of any claims univocally accepted by the mathematical community—modal nominalist

theorizing must avoid interfering with mathematical practice. However, it is not a priori

given that modal nominalism does not interfere with mathematical practice. In fact, an

argument to the effect that nominalism does interfere with mathematical practice—one that

does not rely on any specious appeal to general theoretical virtues—can be distilled from

Penelope Maddy’s interpretation of naturalism. My aim in this chapter is to show that the

naturalistic resources embraced by Maddy’s naturalism are not capable of being used to

show that modal nominalism is in any kind of malevolent conflict with mathematics.

Maddy’s prefers to use the label “Second Philosophy” to denote her naturalism.

The term “Second Philosophy” is used in contrast with “first philosophy.” The first-

philosophical perspective treats philosophical inquiry as an autonomous enterprise, one

that contains its own norms and ideals, and one that is to be conducted using its own

methods and standards of evidence (e.g., Descartes’ method of doubt). Instead, the Second-

Philosophical perspective holds that philosophical inquiry is only legitimate when it arises

from within the broad and open-ended enterprise of empirical science. A salient item


Reflections on Maddy 235 Introduction

here is Maddy’s rejection of a priori knowledge, because such knowledge is not clearly

generated by the use of the methods and evidential standards of empirical science. In

contrast with Burgess’s broadly Quinean understanding of scientific method, the Second

Philosopher aims for discipline-specific engagement with the methodologies of the various

disciplines of science. Maddy finds it particularly striking that although mathematics is

extremely useful in empirical science, nevertheless much of mathematics is pursued as an

autonomous discipline—thus the Second Philosopher is compelled to come to terms with

the actual methods adopted by practicing mathematicians. For Maddy, questions about

the metaphysics of mathematics (to the extent that they are naturalistically legitimate

questions), including questions about the existence or not of mathematical objects, are to

be referred first (and perhaps only) to the methods of mathematics.

I begin in section two by providing an overview of the basic naturalistic resources Sec-

ond Philosophy employs in its assessment of mathematics. The Second Philosopher insists

that mathematics is to be understood and evaluated on its own terms. To say that math-

ematics must be evaluated on its own terms means, roughly, that mathematical methods

and results are not to be criticized on extramathematical grounds. To say that mathematics

must be understood on its own terms means, roughly, that a characterization of the methods

and subject-matter of the discipline must be conducted using only the methods of math-

ematics. How do these naturalistic scruples enter into Maddy’s own assessment of the

methodology and metaphysics of mathematics? Maddy argues that mathematical practice

(in particular, the pursuit of new axioms for set theory) is constrained by objective facts of

mathematical depth—facts about what makes for fruitful and promising avenues of math-

ematical research. She posits these facts as the underlying reality of mathematics, and they

in turn function as reins on metaphysical and philosophical theorizing about mathematics:

For example, metaphysical theorizing about the nature and existence of mathematical

objects is only in good Second-Philosophical standing when such theorizing is compatible

with the idea that purely mathematical forms of justification are sufficient for revealing



the nature and existence of mathematical objects (as described under the hypothesized

metaphysical theory). In other words, metaphysical theorizing about mathematics is to

be rejected if such theorizing introduces a justificatory gap between mathematics and its

metaphysics.

Section two closes with the construction of two provisional objections to modal nomi-

nalism. Both objections stress different aspects of the apparent fact that modal nominalism

violates the Second Philosopher’s entreaty to understand mathematics on its own terms.

On the one hand, it is not a built-in feature of modal nominalism that mathematical

forms of justification suffice for justifying the modal claims modal nominalists make about

mathematics. On the other hand, the construction and advancement of modal nominalist

views is not conducted using only the methods of mathematics but instead appeals to

explicitly philosophical resources and forms of motivation. The remainder of the chapter

is given over to providing answers to the following questions: How damaging are these

objections to modal nominalism? Do they show that modal nominalism is incompatible

with Second Philosophy? Do they show that modal nominalism is incompatible with other,

weaker forms of naturalism? And do they show that modal nominalism is in any kind of

malevolent conflict with mathematics or its practice?

In section three I consider the objection that modal nominalism does not confirm

the idea that mathematical forms of justification suffice for justifying the modal claims

modal nominalists make about mathematics. Unfortunately Maddy has not directly

addressed this issue in the course of the construction and elaboration of Second Philosophy.

Nevertheless how she would compartmentalize the transgression the modal nominalist

makes can be gleaned from the objections she has raised against other views. I argue,

through an analysis of Maddy’s own examples, that it is not inherently objectionable to

produce a metaphysical account of mathematics that merely fails to uphold or incorporate

the idea that mathematical forms of justification suffice for justifying metaphysical claims

about mathematics. Metaphysical accounts of mathematics are only objectionable when



they straightaway preclude the possibility that mathematical forms of justification suffice

for justifying metaphysical claims about mathematics. Ultimately the concern here is not

that such metaphysical accounts violate Maddy’s entreaty to understand mathematics on

its own terms; instead the concern is that such metaphysical accounts violate Maddy’s

entreaty to evaluate mathematics on its own terms. I argue that modal nominalism, at

least in certain of its formulations, is such that it does not preclude the possibility of using

mathematical forms of justification to justify the modal claims modal nominalists make

about mathematics, and moreover that modal nominalism can be coherently amended

so as to accommodate Maddy’s claim that considerations of depth or fruitfulness play an

important evidential role in mathematics. Thus, modal nominalism does not engender any

kind of conflict with mathematical practice.

In section four I consider the objection that modal nominalism cannot be described

or motivated using only the methods of mathematics. In order to understand what is

purportedly beneficial about views that are capable of being described and motivated

using only the methods of mathematics, I discuss the two views Maddy takes to have

inherent affinities with mathematical practice—Thin Realism and Arealism. Maddy argues

that there is no substantive difference between Thin Realism and Arealism, and this

argument takes advantage of the claim that the facts of mathematical depth are the facts

that matter when it comes to appraising accounts of mathematics. Thus, it is a hypostasized

feature of Second Philosophy that theorizing about mathematics is only legitimate when

such theorizing is sanctioned or contained by mathematical methods. Therefore modal

nominalism is not compatible with Second Philosophy. Is the incompatibility of modal

nominalism with Second Philosophy a reason to reject modal nominalism? In sections five

and six I make a provisional case for answering this question in the negative.

In section five I question the overall coherence of Maddy’s naturalism, given that it

requires a rather strong interpretation of what it means to understand mathematics on

its own terms. I am only able to find one genus of evidence in support of the idea that



metaphysical accounts of mathematics must be contained by the methods of mathematics.

This is that using extramathematical resources in the construction and advancement

of metaphysical accounts of mathematics places the metaphysicist at an increased risk

of rejecting mathematical results and methods on non-mathematical grounds. But, as

I shall have argued in previous sections, this risk does not inhere in all metaphysical

accounts of mathematics. In particular, it does not inhere in all modal nominalist accounts

of mathematics. That Maddy nevertheless continues to object to all extramathematical

methods seems to me to be evidence that Maddy’s entreaty to understand mathematics on

its own terms is itself a philosophical claim, as opposed to a mathematical claim. As such,

this entreaty is either self-undermining (if posed as a claim about mathematics and its

methods), or special-pleading (if posed as a philosophical claim about mathematics). Either

way, this component of Second Philosophy appears to be implausibly strong. Why is the

Second Philosopher permitted to make this philosophical claim about mathematics, while

at the same time modal nominalists are not permitted to make the kinds of philosophical

claims about mathematics they wish to advance?

Section six closes the chapter and dissertation with a brief discussion about the com-

patibility of modal nominalism with weaker forms of naturalism. I argue that modal

nominalism is compatible with forms of naturalism that embrace the entreaty to evaluate

mathematics on its own terms, and further that modal nominalism is compatible with

forms of naturalism that seek to find a place for the methodology of mathematics alongside

the metaphysics of mathematics. Modal nominalism is only incompatible with Maddy’s

particularly strong version of naturalism, which unnecessarily runs together a study of the

methods of mathematics and a study of the metaphysics of mathematics.

But what precisely does Second Philosophy have to say about mathematics? Why

in particular does Maddy run together mathematical methodology and mathematical

metaphysics? And what might this mean for modal nominalism? Let me turn now to these

issues.


Reflections on Maddy 239 Second Philosophy of Mathematics

5.2 Second Philosophy of Mathematics

Second Philosophy embodies Maddy’s characterization of the “fundamental naturalistic

impulse,” which is, “the conviction that a successful practice should be understood and

evaluated on its own terms” (Maddy 1997, 201). For Maddy’s naturalist, then, the only

legitimate methods are those that come from successful practices. Though she does not

offer the notion of a “successful practice” up for analysis, she intends the term to refer to

the set of open-ended, constantly evolving, and well-confirmed methods used in empirical

science (Maddy 2011, 39). Science therefore is not to be trusted because it is science; rather,

science is to be trusted to the extent that it employs the best of the available evidential

practices.1

The Second Philosopher becomes interested in mathematics because it plays a vital

role in many of her best scientific theories and evidential practices. But at this point the

Second Philosopher makes an observation that would be somewhat alien to her Quinean

predecessors—she sees that mathematics is practiced autonomously, using methods quite

different from those used in empirical science.2 Mathematics is thereby to be counted as a

successful discipline in its own right, and consequently compels the Second Philosopher

to understand and evaluate mathematics on its own terms.

Unfortunately, Maddy does not explain in much detail, at least in a general way, what

it means to understand and evaluate mathematics on its own terms, but instead relies on

numerous examples and case studies that she takes to exemplify the kinds of naturalistic

scruples with which the Second Philosopher identifies. For this reason I am not certain that

it is possible to fully understand and appreciate Maddy’s naturalism without surveying

the specific examples and case studies she discusses. Nonetheless it should be helpful to

1Some take this to be a glaring defect—what prevents a theologian or an astrologer from reckoning theirevidential practices as producing successes in their own disciplines? What ultimately distinguishes themethods of these disciplines from the “authoritative” methods used in empirical science? For this kind ofcriticism, see (Dieterle 1999), (Marfori 2012), (Rosen 1999), and (Weir 2005). For a response, see (Maddy2007,345-7) and (Tappenden 2001).

2Maddy further argues that the autonomy of mathematics is an important aspect of its usefulness toscience (2007, 330-1).



identify and describe up front what kinds of naturalistic scruples are embraced by the

Second Philosopher. In doing this I ask the reader’s indulgence—Second Philosophy is a

particularly strong version of naturalism and it may not be initially obvious why Maddy

adopts the stronger of the principles described below.

Consider first Maddy’s entreaty that mathematics be evaluated on its own terms. Es-

sentially what this means is that mathematical results and forms of reasoning are not to

be criticized on extramathematical grounds—sound mathematical results and methods,

once established, are immune to criticism from outside of mathematics. To use a simplistic

example, it would be a failure to evaluate mathematics on its own terms if one were to

reject Euclid’s proof of the existence of infinitely many prime numbers solely on the basis

of the prior conviction that mathematical objects do not exist. It would also be a failure to

evaluate mathematics on its own terms to reject non-constructive forms of mathematical

reasoning on the basis of the kinds of metaphysical and epistemological concerns that

motivate intuitionists.3 The Second Philosopher, then, objects to all accounts of mathemat-

ics that involve the rejection, on non-mathematical grounds, of established mathematical

results and methods. Let me label these two transgressions as “result-rejecting” and

“method-rejecting,” respectively. Avoiding both method- and result-rejecting is a necessary,

but not sufficient, condition for a philosophical account of mathematics to be consistent

with Second Philosophy.

Consider next Maddy’s entreaty that mathematics be understood on its own terms. The

basic idea here is that various issues related to mathematics and its practice, including an

account of the subject-matter of mathematics, are to be identified and described using only

the methods of mathematics. But Maddy’s naturalism does not only place reins on the

admissible methods for conducting inquiries about mathematics, it also places restrictions

on the legitimate subjects of inquiry. Roughly, a subject of inquiry is only legitimate when

3It would not, of course, be a failure to evaluate mathematics on its own terms to pursue intuitionisticmathematics as a form of mathematics in itself—the problem is with those who seek to replace classicalmathematics with intuitionistic mathematics.



its investigation facilitates the realization of some identifiable mathematical goal. If there

are no perspicuous mathematical reasons for investigating X , then the Second Philosopher

denies that there is any motivation for or wisdom in investigating X .4 I will use the term

“method-contained” as a label for accounts of mathematics that embrace Maddy’s entreaty

that mathematical methods are to place firm restrictions on the admissible methods and

subjects of inquiry for investigating mathematics.

There is a weaker notion that Maddy sometimes invokes when attempting to “un-

derstand” mathematics on its own terms. This is the idea that a required feature of

philosophical accounts of mathematics is that they either actively pursue or leave open

the possibility of explaining how it is that mathematical forms of justification—which,

Maddy claims, are sufficient for establishing mathematical results and are innocent of

any philosophical prejudices—are furthermore sufficient for generating reliable beliefs

about the underlying reality of mathematics. In other words, it must be possible to show,

without using extramathematical resources, that the mathematical facts—according to

the philosophical accounting of these facts—are in genuine agreement with established

mathematical results. For instance, if a philosophical proposal says that mathematics de-

scribes a realm of abstract mathematical objects, then this proposal is only compatible with

Maddy’s naturalism if it is plausible that mathematical forms of evidence—unsupplemented

by philosophical views about, e.g., a priori knowledge—are sufficient for justifying claims

about abstract mathematical objects. I will use the term “method-affirming” to describe

philosophical accounts of mathematics that are consistent with Maddy’s entreaty that

mathematical forms of evidence are sufficient in the way just described.

Though it is reasonably clear that avoiding both method- and result-rejecting are nec-

essary conditions for compatibility with Maddy’s naturalism, it is less clear that being

both method-affirming and method-contained are necessary conditions for compatibility

with Maddy’s naturalism. Perhaps these latter two are sufficient, but not necessary, condi-

4Unless, of course, X is relevant to some other naturalistically legitimate area of inquiry, e.g., chemistryor physics.



tions for being consistent with Maddy’s naturalism. What is of primary interest here is

how these four conditions can be used to support or criticize philosophical accounts of

mathematics, and, in particular, how these conditions might be used to support or criticize

modal nominalist accounts of mathematics. In a moment I shall lodge the provisional

Second-Philosophical objection that modal nominalism fails to be both method-affirming

and method-contained. The bulk of this chapter will be given over to understanding just

how objectionable (or not) modal nominalism is on account of its apparent failures to be

both method-affirming and method-contained. But before developing this criticism, it

should be worthwhile to examine Maddy’s own outlook on the metaphysics of mathemat-

ics in closer detail. Doing this should help shed further light on what is involved in being

method-contained and method-affirming, and will provide important background infor-

mation in support of the claim that modal nominalism fails to be both method-affirming

and method-contained.

Maddy’s particular focus within mathematics has been on the methodology of the

pursuit of new axioms for set theory. A persistent feature of this work involves the

development of an account of mathematics that is what I have described as being method-

contained and method-affirming. For Maddy, whatever actually constrains or influences

mathematical work should be accorded the status of the underlying reality of mathematics.

And based on her observations of the work of leading set theorists, what actually constrains

mathematical practice is not, contrary to traditional platonist accounts of mathematics,

some independently-existing realm of timeless or eternal abstract mathematical objects, but

instead a collection of objective facts about what makes for deep or fruitful mathematics—

the facts of mathematical depth:5

. . . this account of the objective underpinning of mathematics—the phenomenonof mathematical fruitfulness—is closer to the actual constraint experienced bymathematicians than any sense of ontology, epistemology or semantics; whatpresents itself to them is the depth, the importance, the illumination providedby a given mathematical concept, theorem, or method. A mathematician may

5Thanks to Susan Vineberg for suggesting this way of framing Maddy’s project.



blanch and stammer, unsure of himself, when confronted with questions oftruth and existence, but on judgments of mathematical importance and depthhe brims with conviction. For this reason alone, a philosophical position thatputs this notion center stage should be worthy of our attention. (Maddy 2011,116-7)6

Here Maddy trades on the idea that since mathematics is an autonomous discipline the

relevant evidence for or against a mathematical claim never involves exterior philosophical

input but is instead all and only mathematical evidence. One of her central claims is

that considerations of fruitfulness play an important evidential role in mathematics—

e.g., the acceptability of a new axiom or concept is a function of whether the proposed

axiom or concept advances the practice in a tangible, objective way. Moreover, proper

advancement consists not in realizing arbitrary or merely fashionable mathematical goals

(Maddy 2011, 81), rather, “mathematical goals are only proper insofar as satisfying them

furthers our grasp of the underlying strains of mathematical fruitfulness” (ibid., 82).

Clearly, then, Maddy views judgments about fruitfulness and importance as part of

sound mathematical methodology. This methodology can thereby function as a guide to

mathematical metaphysics—the underlying reality of mathematics is given by the facts

of mathematical depth, and one important function of mathematical methodology is to

reveal these facts.7

Maddy provides the group concept as one example of a mathematically deep concept:

In the logical neighborhood of any central mathematical concept, say the con-cept of a group, there are innumerable alternatives and slight alterations thatsimply aren’t comparable in their mathematical importance. . . ‘group’ standsout from the crowd as getting at the important similarities between structuresin widely differing areas of mathematics and allowing those similarities to bedeveloped into a rich and fruitful theory. . . ‘group’ effectively opens the doorto deep mathematics in ways the others don’t. So what guides our conceptformation, beyond the logical requirement of consistency, is the way somelogically possible concepts track deep mathematical strains that the others miss.

6Cf. David Corfield’s realism about ‘inherent structure’ (2003, 31).7Thanks again to Susan Vineberg for suggesting this way of presenting Maddy’s views on methodology

and metaphysics.



(ibid., 79)8

Similar considerations are mustered in support of various new axiom candidates for set

theory, including the historical case for the Axiom of Choice, and the more contemporary

cases for several of the (less than full-blown) determinacy axioms that are thought to be

consistent with Choice (Maddy 2011, 80-1).

It is difficult to say precisely what kinds of claims that considerations of depth or

fruitfulness are alleged to justify or support. Does the fact that the group concept is

mathematically deep justify the truth simpliciter of claims about groups, including existence

claims? Or does the fact that the group concept is mathematically deep justify only the

more pragmatic claim that studying groups helps to realize various kinds of mathematical

goals? Or something else entirely? As I shall explain in more detail later in the chapter,

Maddy’s official position is that answering these questions falls outside the province

of mathematics, and given her advocacy of method-contained accounts of mathematics,

there is consequently no fact of the matter about how they should be answered. On her

analysis, all that mathematical methods strictly license are depth claims and mathematical

claims unadorned by any kind of philosophical interpretation. (I should note here that

Maddy denies that mathematical methods positively support the idea that mathematics is

comprised of a body of truths. More on this and related issues in §4 and in §5.)

In any case, one might worry that Maddy’s observations about the role of fruitfulness in

concept formation and axiom selection do not generalize and arise because of idiosyncratic

features of set theory and group theory. Some have indeed complained that Maddy’s

approach to methodology is unduly influenced by her focus on set theory and that this

approach risks becoming itself irrelevant to the rest of mathematics.9 However there

is evidence that fruitfulness has a role to play in mathematical concept formation more

generally. For instance, Jamie Tappenden points to several examples in number theory and

abstract algebra (2008a, 2008b); meanwhile, David Corfield offers examples from algebraic8For a similar observation about the group concept, see (Corfield 2003, 30).9See, e.g., (Riskin 1994) and (Decock 2002).


Reflections on Maddy 245 Two Objections to Modal Nominalism

topology (2003). If Tappenden and Corfield are correct, Maddy’s approach to methodology

in set theory—that proper method is constrained by considerations of mathematical

fruitfulness—is likely to have exemplifications throughout the various subdisciplines of

mathematics. Thus the claim that mathematical methodology (along with the purported

metaphysical implications of this methodology) responds to fruitfulness considerations is

unlikely to constitute a violation of any of the four conditions described above.

5.2.1 Two Provisional Second-Philosophical Objections to Modal Nominalism

Identifying the various naturalistic principles at work in Second Philosophy provides a

means for determining how alternative accounts of mathematics can come into conflict

with Maddy’s naturalism. If modal nominalism is inconsistent with Second Philosophy,

then it stands to reason that this is because modal nominalism is either method- or result-

rejecting, or else modal nominalism either fails to be method-affirming or fails to be

method-contained. Here I raise two objections to modal nominalism. First, that modal

nominalism is not method-affirming, and second, that modal nominalism is not method-

contained. Both objections are offered as evidence that modal nominalism is incompatible

with Second Philosophy. The extent to which these objections are damaging to modal

nominalism will be explored in the remaining sections of the chapter.

I think it is important to recall that my project in this dissertation—to defend the modal

nominalist approach in philosophy of mathematics—is not to decisively establish the

major nominalist thesis that mathematical objects do not exist. Rather, my aim is to show

that modal nominalist theories are coherent and are capable of overcoming the kinds of

objections that Stewart Shapiro, Burgess, Maddy, and others raise (or might raise) against

them. (I do of course hope that, along the way, I have offered, and will continue to offer,

reasons for supposing that modal nominalist theories are preferable to the competition.) It

seems unlikely that the major thesis of nominalism can be established without substantive

appeal to simplicity principles like Ockham’s razor, but I reject the idea that simplicity

principles are required for advocating modal nominalist theories—theories that are consistent



with, but do not require, the non-existence of mathematical objects. Other motivations are

available for modal nominalist theories.

Now the alternate motivation I have proposed at various times in this dissertation—

restraining from making existence assertions without proper evidence—does seem of

a piece of the general idea, much a part of any plausible form of naturalism, that one

should only make assertions when one’s evidence adequately supports one’s assertions.

Indeed, Maddy’s own case studies suggest, contra Quine’s Indispensability Argument

for the existence of mathematical objects, that the indispensable inclusion of a posit in a

scientific theory that is in possession of the “scientific virtues” is not by itself the right

kind of evidence for drawing the conclusion that the posited object exists (Maddy 1997,

133-57). For instance, scientific idealizations, including the continuity of spacetime, may

perform an indispensable role in scientific theorizing, nevertheless scientists consider it

an open question as to whether spacetime is actually continuous (ibid., 151-2). Similarly,

the atomic theory did not gain widespread acceptance in the scientific community until

Perrin’s experiments provided it with decisive observational support, even though prior

to these experiments it could have been argued that the atomic theory was in possession

of the “scientific virtues” (ibid., 135-43). So my sympathy for modal nominalist theories is

analogous to Maddy’s position concerning scientific idealizations. Further, Maddy herself

acknowledges that mathematical methods do not support robust answers to questions

about the existence or not of mathematical objects. This confirms an oft-voiced claim of this

dissertation, viz., that it is no presumption of mathematics proper that mathematical objects

exist. This suggests a somewhat more explicit motivation for modal nominalist theories—

that they facilitate the goal of providing an account of the content of mathematical theories

without making such a presumption either.

Nevertheless it is unclear that this manner of motivating the modal nominalist ap-

proach is capable of positively supporting any of the particular metaphysical claims modal

nominalists make about mathematics. Modal nominalists, even though they need not



appeal to Ockhamite considerations, nevertheless appeal to more than just the methodology

of mathematics in the construction and advancement of their views. For instance, Hartry

Field imposes the judgment that mathematical knowledge is a combination of modal and

logical knowledge (1989c). Meanwhile Geoffrey Hellman suggests that, “mathematics is

the free exploration of structural possibilities, pursued by (more or less) rigorous deductive means”

(1989, 6). But, on the Second Philosopher’s analysis, mathematical methods themselves

support only very limited metaphysical claims about the content of mathematics, viz.,

claims about the fruitfulness or depth of various axioms and concepts. Thus, under Second

Philosophy, the underlying reality of mathematics consists neither in the truths of logic nor

in the facts about structural possibilities, but instead in the facts of mathematical depth. It

follows that Field’s fictionalism and Hellman’s Modal Structuralism are not endemic to the

methods of mathematics. Therefore both fictionalism and Modal Structuralism fail to be

method-contained, and therefore appear to be incompatible with Second Philosophy.

Unlike Field and Hellman, Charles Chihara’s Constructibility Theory does not provide

an account of the underlying reality of mathematics but instead offers a framework for car-

rying out mathematical reasoning in a way that (a) treats mathematics as a body of truths,

and (b) is not committed to the existence of mathematical objects.10 Still, it is unclear why a

Second Philosopher should be moved to reconstruct mathematical reasoning so as to avoid

reference to mathematical objects. If it is correct that the existence of mathematical objects

is no presumption of mathematics but is instead a certain kind of philosopher’s presumption

about mathematics, then the impulse to combat this philosophical presumption is not a

source of motivation for Constructibility Theory that is endemic to the methods of mathe-

matics. Thus, Constructibility Theory itself is not endemic to mathematical methodology.

Therefore, Constructibility fails to be method-contained and consequently appears to be

incompatible with Second Philosophy.

All three of these views also fail to be method-affirming. It is not a built-in feature of ei-

10Of course, Hellman is also attempting (a) and (b), and Field (b), but each is doing more besides—offeringan account of the underlying reality of mathematics.



ther Modal Structuralism, fictionalism, or Constructibility Theory that mathematical forms

of justification—which are, allegedly, responsive to the facts of mathematical depth—are

sufficient for justifying the content of the modal nominalist interpretations of mathematical

claims. It is not clear, for instance, why the alleged fact of the fruitfulness of the ZFC axioms

is thereby evidence that ZFC is primitively logically possible (in Field’s and Hellman’s

cases), or is thereby evidence for the constructibility claims that Chihara would advance

in place of the ZFC axioms. That modal nominalism fails to be method-affirming, then,

presents a further ostensible reason for supposing that modal nominalism is incompatible

with Second Philosophy.

That modal nominalism fails to be both method-contained and method-affirming forms

the basis for two arguments which purport to unveil modal nominalism as incompatible

with Second Philosophy. But how damaging are these conclusions? My goal in the

remainder of the chapter is to chart the extent of the damages by determining how the

Second Philosopher would answer the following questions, and whether the Second

Philosopher is justified in giving the answers that she would in fact give: Is the alleged

incompatibility of modal nominalism with Second Philosophy a mere flesh wound or

is it evidence of an underlying, mortal defect of modal nominalism? In particular, are

the purported facts that modal nominalism fails to be method-contained and fails to

be method-affirming evidence that modal nominalism not only conflicts with Second

Philosophy, but also with mathematics itself? That is, are these facts evidence that modal

nominalism is also method- and result-rejecting? Unfortunately, Maddy nowhere directly

addresses these questions. She does however address similar questions regarding various

alternative accounts of mathematics. So before I can address the precise nature of the

difficulties (or lack thereof) that modal nominalism faces on account of its purported

failure to be both method-contained and method-affirming, it is first necessary to consider

Maddy’s actual uses these naturalistic scruples in judgment against alternative accounts of

mathematics.


Reflections on Maddy 249 The Method-Affirming Objection

In the next section I use Maddy’s objections to competing accounts of mathematics to

determine whether accounts of mathematics that fail to be method-affirming are thereby

genuinely incompatible with Second Philosophy, and also to determine what other kinds

of objections might be raised against such accounts. What I find is that Maddy’s decisive

objection to most competing accounts of mathematics is not that they fail to be method-

affirming, but instead that they are either method- or result-rejecting in a malevolent way.

(N.B., there are benevolent forms of method- and result-rejecting.) It is possible for an

account of mathematics to fail to be method-affirming and yet avoid both method- and

result-rejecting. In §4 I use Maddy’s discussion of her favored views—Thin Realism

and Arealism—to show that being method-contained is a genuine requirement as far as

consistency with Second Philosophy is concerned, but that accounts of mathematics that

fail to be method-contained are not for that reason either method- or result-rejecting. These

developments are used in §5 to provide a more general appraisal of Maddy’s naturalism,

and are also used in §6 to determine the compatibility of modal nominalism with various

forms of naturalism that relax some of Maddy’s stronger naturalistic principles.

5.3 The Method-Affirming Objection to Modal Nominalism

Given that modal nominalism fails to be method-affirming, should this count as a decisive

reason to reject modal nominalism, even from the Second-Philosophical perspective? Why

should being method-affirming be considered a virtue for accounts of mathematics? In

this section I argue that being method-affirming is virtuous primarily for its prophylactic

qualities—method-affirming views run very little risk of method- and result-rejecting.

In order to appreciate this point it is necessary to examine some of Maddy’s objections

to various views that fail to be method-affirming—this should function as an aide to

understanding the ways in which accounts of mathematics do and do not come to be

method- and result-rejecting.


Reflections on Maddy 250 A Miscellany of Objections

5.3.1 A Miscellany of Objections

Consider Maddy’s analysis of the array of positions she refers to by the blanket term

‘Robust Realism.’ The “defining feature” of Robust Realist theories is that they view math-

ematics as, “the study of some objective, independent reality” and that they analyze the

justification for beliefs about this reality as arising other than trivially from sound mathe-

matical reasoning (Maddy 2007, 365). This reality—the entities about which such theories

profess realism—could be comprised of abstract mathematical objects, of mathematical

structures, of modal facts, or of something else entirely (Maddy 2011, 56-7). The basic idea

unifying Robust Realist theories is that purely mathematical reasons are not sufficient for

justifying mathematical assertions, because the existence or not of mathematical objects

or structures (or the obtaining or not of modal facts) is independent from mathematical

theorizing—theorizing that, according to Maddy, tracks facts about what makes for deep

or fruitful mathematics. It should be obvious that Robust Realism thereby fails to be

method-affirming (in addition to failing to be method-contained). Interest here is in what

is objectionable about Robust Realism in virtue of its failing to be method-affirming. Maddy

argues that Robust Realism consequently faces two related sets of difficulties.

The first problem is the creation of epistemological difficulties. If mathematical methods

are not sufficient for justifying claims about the underlying reality of mathematics, then

the source of knowledge about this underlying reality must come from somewhere other

than the methods of mathematics. Consider traditional platonism. What the platonist

requires is some independent reason for thinking that the mathematical realm is genuinely in

agreement with the accepted assertions of mathematics. But it is doubtful that the platonist

can provide such reasons without appealing to naturalistically occult faculties (e.g., a

priori forms of justification). Thus the platonist seems unable to justify, on naturalistically

acceptable grounds, that mathematical objects exist, making it something of a mystery how

anyone is supposed to justify assertions that refer to these objects. The general moral for

Robust Realism is that since there is a gulf between mathematical methods and the Robust



Realist’s underlying reality, “we have no way of ruling out the possibility that reality is

sadly uncooperative, that [in the case of set theory] much as we’d like to use sets in our

mathematical pursuits, they just don’t happen to exist” (Maddy 2011, 58). Nevertheless

this does not show that the Robust Realist is, in principle, in material disagreement with

mathematics—the Robust Realist might be lucky to defend a metaphysical position that

perfectly matches the accepted assertions of mathematics. But Maddy thinks there would

be something awry even if the Robust Realist did wind up in material agreement with

practice, which brings me to the second problem.

The second problem is that there is something untenable about the Robust Realist’s

initial isolation from method (even prior to the creation of the aforementioned epistemo-

logical difficulties). And this is that the very notion that mathematical assertions require

non-mathematical supplementation is problematic under Second Philosophy:

But if the Robust Realist is right, if the goal of set theory is to describe anindependently-existing reality of some kind, then it appears that Cantor’sevidence needs supplementation, and not supplementation of the same sort,like adding in Dedekind’s grounds and so on, but supplementation of anentirely different kind: we need an account of how the fact that sets serve thisor that particular mathematical goal makes it more likely that they exist. . . Tothe Second Philosopher, this hesitation seems misplaced: why should perfectlysound mathematical reasoning require supplementation? Hasn’t somethinggone wrong when rational mathematical methods are called into question inthis way? (ibid., 58)

Maddy’s assessment here is that, for the Robust Realist, mathematical evidence is not

good enough, all things considered. In other words, the Robust Realist maintains that

mathematical claims must be evaluated on extramathematical grounds. However, according

to the Second Philosopher, mathematical methods are internally sufficient for justifying

mathematical assertions. This is a feature of mathematical practice that Robust Realists

do not simply fail to affirm but that they allegedly reject. Thus, Robust Realism is method-



rejecting, beyond its evident failure to be method-affirming.11,12

Consider next Maddy’s objection to Quine’s indispensability-style realism about math-

ematical objects. Quine’s view can be described as a form of Robust Realism in the sense

that Quine elicits non-mathematical justification in order to describe the underlying reality

of mathematics—which mathematical objects exist depend on what bits of mathemat-

ics get applied in empirical science. This means that Quine’s realism threatens to be

method-rejecting for the same reasons as Robust Realist views in general threaten to be

method-rejecting: purely mathematical considerations are not given primary authority

when it comes to justifying mathematical assertions.13 Nevertheless the Quinean realist

(at least, Quine himself) commits a further transgression: Quine argues for the Axiom of

Constructibility (V = L) as an extension of ZFC because it is an economical cut off point

for the set-theoretic hierarchy—allegedly, V = L can accommodate all of the mathematics

that is needed for science.14 Maddy’s detailed examination of set theory, however, suggests

11Maddy describes Shapiro’s structuralism, Hellman’s Modal Structuralism, and Godel’s platonism eachas a form of Robust Realism (ibid., 56-7). These analyses may prove problematic. Godel seems willing toallow depth considerations a role in justifying mathematical axioms (though his views on mathematicalintuition seem to support Maddy’s description of him as a Robust Realist). Hellman does not attempt toconstruct an epistemology for his primitive modal claims, but I think he is not appropriately described as aRobust Realist for reasons I will give later in the chapter (at least, if Hellman is a Robust Realist, then I amnot convinced that the objections presented here apply to all Robust Realist views). Shapiro’s coherence-stylejustifications for the existence of structures (see chapter two) is sufficiently non-trivial for the Robust Realistlabel to stick (a situation not helped by Shapiro’s pattern-based epistemology). However, it is not clearthat ante rem structuralism is method-rejecting, because it defers (perhaps controversially) to mathematics inorder to determine what structures are coherent, and hence, what structures exist. It is an open questionwhat views are genuinely Robust Realist in the sense of being method-rejecting, and hence in conflict withpractice—so there is a possibility of using my defense of modal nominalism as a springboard for a defense ofother kinds of philosophical accounts of mathematics.

12Similar reasoning is alleged to defeat the neo-Fregeanism espoused by Crispin Wright. According toMaddy, Wright’s argument for thinking that set theory is truth-apt is that, “a minimalist truth predicate canbe defined for any such discourse in such a way that statements assertable by its standards come out true,”and that set theory, “enjoys certain syntactic resources and displays well-established standards of assertion”(ibid., 70). Thus, a minimalist truth predicate can be defined for set theory. But, “[i]n contrast, the Thin Realisttake [sic] set theory to be a body of truths, not because of some general syntactic and structural featuresit shares with other discourses, but because of its particular relations with the defining empirical inquiryfrom which she begins” (ibid.). The objection, then, is that Wright’s strategy for justifying the assertoricstatus of set-theoretic claims is in conflict with the strategy implemented by the Second Philosopher (inparticular, the Second Philosopher qua Thin Realist). Wright’s minimalist account of mathematical truth,then, is method-rejecting.

13And so Quine’s realism is also trivially neither method-contained nor method-affirming.14On this score, Quine writes that, “[s]o much of mathematics as is wanted for use in empirical science

is for me on a par with the rest of science. Transfinite ramifications are on the same footing insofar as they



that there are sound mathematical reasons for rejecting V = L in favor of much more

expansive extensions (e.g., determinacy axioms that are consistent with choice). Thus,

Quine’s realism seeks to reject, on extramathematical grounds, a group of mathematical

results or assertions that are nevertheless mathematically supported. Quine’s realism, then,

commits the further transgression of being result-rejecting.

Both method- and result-rejecting are potential factors in Maddy’s assessment of fic-

tionalism:

The central challenge is to delineate and defend the proper ways of extendingwhat the fictionalist calls the ‘set theoretic story’, but calling it that, ratherthan just ‘set theory’, doesn’t appear to advance our understanding on thispoint. . . the value of the fictionalist analogy is limited, and keeping it at theforefront of our thinking about set theory might tempt us to impose categoriesand judgments foreign to our subject and to ignore important features withoutcorrelates in fiction. (Maddy 2011, 98)

Note the entreaty to enact tariffs against fictionalism—treating mathematics as an elaborate

fiction risks imposing foreign (i.e., extramathematical) judgments on mathematics, which

would unveil fictionalism as result-rejecting or method-rejecting, depending on the nature

of these judgments. For instance, the blanket judgment that Field makes—that all mathe-

matical existence assertions are false—could be construed as result-rejecting. Alternatively,

to insist (as a fictionalist might) that certain fictionalist notions, e.g., truth-in-a-story, apply

in the assessment of mathematical assertions, could be construed as method-rejecting.

At this point it would seem that the more (and perhaps the only) decisive objection on

offer for the views just entertained is not that they fail to be method-affirming, but rather

that they engage in either method- or result-rejecting. This raises two questions. The first

question concerns the substratal virtue of method-affirming accounts of mathematics—

perhaps being method-affirming is a mere prophylactic against being either method-

or result-rejecting. After all, merely failing to support a method is not equivalent to

rejecting a method. Are there nevertheless substantive examples in which a failure to

come of a simplificatory rounding out, but anything further is on a par rather with uninterpreted systems”(1984, 788).


Reflections on Maddy 254 Method-Affirming as a Prophylactic

be method-affirming by itself is responsible for some kind of objectionable conflict with

mathematics? The second question concerns whether being method- and result-rejecting

are the objectionable qualities that Maddy takes them to be. Are all forms of method- and

result-rejecting naturalistically objectionable? Or are there benevolent forms of method-

and result-rejecting? Let me address these questions in turn.

5.3.2 Method-Affirming as a Prophylactic

The failure to be method-affirming appears to factor prominently in Maddy’s criticisms

of if-thenist and Carnapian-framework-style accounts of mathematics. In reference to

the suggestion that various axiomatizations of set theory merely constitute alternative

set-theoretic “frameworks”:

The trouble with this suggestion is that it fails to capture one of the elements ofset-theoretic practice we’re most eager too describe and assess: the addition ofnew axioms. On this view, one axiom wouldn’t be selected over another forcompelling set-theoretic reasons—these are all internal to the framework—butas a pragmatic, conventional decision to move from one linguistic frameworkto another. Obviously this is not a suitable path for the Second Philosopher.(ibid., 69)

Two problems are identified in this passage. The first is that the framework-style account

of mathematics “fails to capture” an important element of set-theoretic practice. The

second is that the pragmatic-conventional-style justifications given under the framework-

style account of mathematics undercut “compelling set-theoretic” forms of justification.

The second problem is simply that this account of mathematics is method-rejecting. But

what about the first problem? That the framework-style account of mathematics fails

to capture the methodology of axiom addition implies that the account cannot capture

the idea that mathematical forms of justification—which respond to considerations of

fruitfulness—suffice for justifying mathematical claims. This, then, is an accusation that

the framework-style account of mathematics fails to be method-affirming. But what

precisely is the force of this component of the objection? Is the framework-style account

objectionable simply because it fails to be method-affirming, or because its failure to be



method-affirming is somehow related to its being method-rejecting? A clue perhaps arises

in Maddy’s discussion of if-thenism.

If-thenism, also known as deductivism, is the view that mathematics can be described

as the study of what follows from what. The if-thenist reinterprets all categorical mathe-

matical assertions as conditionals—instead of “the Well-Ordering Theorem is true,” the

if-thenist instead claims that “if the axioms of ZFC are true, then the Well-Ordering Theo-

rem is true,” or that “if the set-theoretic hierarchy exists, then every set can be well-ordered.”

If-thenism is compatible with agnosticism about (or even disavowal of) the truth of the

antecedents of such conditionals. Maddy’s objection to if-thenism, at least in its “crude”

form, is that

. . . though mathematicians are often engaged in proving one thing from another,they obviously don’t regard any starting point, even any consistent startingpoint, as equally worthy of investigation; if one characterizes set-theoreticpractice as that of deriving theorems in one or another axiomatic setting, oneignores the very features of that practice that have been [my] focus, namely, theforces that shape the concepts and assumptions of the setting itself. (ibid., 99)

If-thenism, like the framework-style account, does not capture the methodology of ax-

iom addition. Thus, if-thenism does not support the idea that mathematical forms of

justification—which respond to considerations of fruitfulness—suffice for justifying math-

ematical claims. If-thenism consequently fails to be method-affirming. However, Maddy

does not believe that this is a life-threatening diagnosis in the case of if-thenism. That

if-thenism fails to capture the methodology of axiom addition is only debilitating for those

who offer if-thenism as a complete and non-amendable account of the nature and method-

ology of mathematics. Maddy admits to a degree of sympathy with a more “sophisticated”

if-thenism that admits

. . . that mathematics is more than a matter of determining what follows fromwhat, that mathematicians are also engaged in forming those concepts andselecting those assumptions, and would then assume responsibility for explain-ing how this process is constrained, what principles should guide it and why.(ibid.)



So long as the if-thenist acknowledges her methodological lacunae, her view is not ulti-

mately objectionable (even to the Second Philosopher). It would thus appear that, even

under Second Philosophy, merely failing to include the methodology of axiom selection

is not grounds for dismissal, provided that it is possible to supplement the view under

consideration with an acceptable account of this methodology. So what distinguishes

the if-thenist from the framework-stylist is that latter precludes, but the former does not

preclude, using internal set-theoretic modes of argument to arrive at methodological judg-

ments, where the force of these judgments is more than solely pragmatic. The if-thenist can,

then, acknowledge non-pragmatic, mathematical reasons for or against formulating new

concepts and adopting new axioms. But the framework-stylist denies in principle that

there exist any such non-pragmatic reasons (or at least that any such reasons could be

given within a set-theoretic framework). Though both views fails to be method-affirming,

only the framework-stylist view is thereby also method-rejecting.15

The framework-stylist, then, removes any possibility of supporting the objective exis-

tence of the facts of mathematical depth. Even if Maddy is incorrect to suppose that the

ultimate, underlying reality of mathematics consists of the facts of mathematical depth, one

could still nevertheless agree that there are such facts and that they do provide an objective

constraint on mathematical work in the sense that considerations of fruitfulness play an

important evidential role in mathematical practice. Any view which denies the objective

existence of the facts of mathematical depth would thereby appear to involve some form

of method- or result-rejecting. In other words, views that in-principle prohibit legitimizing

sound method—including the existence of the objective facts of mathematical depth as a

15Can it be objected that Maddy’s account of mathematics is equally pragmatic? I am not certain. Herobjection is that the framework-stylist has it wrong in saying that it is a mere conventional decision whenit comes to selecting an extension of ZFC. On the other hand, Maddy describes the underlying reality ofmathematics as a set of facts about what makes from fruitful mathematics. Is this a substantive distinction?Maddy suspects so—she believes that it is an open question for the framework-stylist which extension ofZFC is most fruitful; for Maddy this is not open—the facts of mathematical depth determine which extensionis most deep. This distinction turns on the assumption that the framework-stylist cannot admit objective factsabout the comparative worth of her available choices concerning extensions of ZFC. It is not obvious to methat the framework-stylist cannot admit such facts, but I do not have space to pursue the matter.


Reflections on Maddy 257 Method- and Result-Rejecting

component of sound method—are guilty of being either method- or result-rejecting. In the

case of such views, although the initial evidence of trouble may come from observing that

they fail to be method-affirming, deep down the problem is that they are method-rejecting.

In the case of framework-style accounts, their failure to be method-affirming serves to

expose covert instances of method-rejecting. Not so with if-thenist accounts. Their failure

to be method-affirming does not arrogate the forthright rejection of if-thenism, but merely

serves to indicate a provisional deficiency. Something of a challenge is presented: For

the if-thenist to explain how her account of mathematics can be supplemented to become

method-affirming, i.e., for the if-thenist to explain how mathematical forms of justification

distinguish the mathematically salient “if”s from the mathematically sterile “if”s.16

5.3.3 Method- and Result-Rejecting: The Good and The Bad

I have thus far taken for granted that result- and method-rejecting are, at least from a

naturalistic perspective, inherently objectionable activities. But it is not obvious that all

method- and result-rejecting accounts of mathematics are necessarily objectionable or in

any kind of malevolent conflict with mathematics and its practice. For instance, Russell’s

discovery of the set-theoretical paradoxes is plausibly described as a result- and method-

rejecting episode in the history of philosophy of mathematics, but the sense in which this

discovery interfered with mathematics was beneficial, rather than malevolent.17

The case of Russell’s discovery of the set-theoretical paradoxes is evidence that the

Second Philosopher and naturalists more generally share an interest in maintaining that

not all forms of philosophical interference with mathematics are bad. What makes Russell’s

method- and result-rejecting acceptable, but, e.g., Quine’s method- and result-rejecting

unacceptable? I suspect that Maddy’s answer would be that Russell’s intervention helped

16It should be noted that Hellman’s Modal Structuralism is a modalized version of if-thenism, which treatsits conditionals (e.g., necessarily, if there exists a model of ZFC, then the Well-Ordering theorem is true inthe model) as non-trivially satisfied by modal-existence postulates (e.g., it is possible for there to exist amodel of ZFC). See below for my thoughts on why Hellman’s position does not preclude the same kind of“sophisticated” amendation that is available to the if-thenist.

17Thanks to Susan Vineberg for suggesting this example.


Reflections on Maddy 258 Method-Affirming Reconsidered

to excise a mathematically fruitless line of research (naıve set theory),18 whereas Quine’s

conflict threatened to excise mathematically fruitful lines of research (large-cardinal and

determinacy axioms). The moral suggested here is that conflicts with practice are only

objectionable when they threaten to direct practitioners away from fruitful avenues of

research—either by excising genuinely fruitful lines of research, or by forcing mathemati-

cians onto fruitless lines of research. This means that method- and result-rejecting accounts,

just like accounts that fail to be method-affirming, are not automatically objectionable. The

only method- and result-rejecting accounts that are objectionable are those that call for

practitioners to abandon or ignore mathematically fruitful research.

There is evidence, then, that the only accounts of mathematics that are naturalistically

and Second-Philosophically objectionable are those that are either method- or result-

rejecting in a malevolent way. The same appears to hold true for accounts of mathematics

that fail to be method-affirming. The above discussion of if-thenism shows that the slope

from failing to be method-affirming to method- and result-rejecting is not a slippery one—it

is possible for an account of mathematics that fails to be method-affirming to nevertheless

avoid malevolent forms of method- and result-rejecting. This suggests an avenue for

defending modal nominalism against the charge that it fails to be method-affirming: To

argue that modal nominalism fails to be method-affirming in a way that does not engender

any malevolent forms of method- or result-rejecting.

5.3.4 The Method-Affirming Objection Reconsidered

Whether modal nominalism’s failure to be method-affirming is evidence that modal

nominalism is ultimately objectionable (to the Second Philosopher or to other naturalists)

depends on whether there is evidence for any of the following claims: That there are clear

obstacles to appending modal nominalism with an account of mathematical method; that

modal nominalism calls for the rejection of mathematical methods or results; and that

18At least, Russell’s intervention helped the mathematical community to realize that the unrestricted com-prehension axiom of naıve set theory violated an already established mathematical precept—the requirementof logical consistency.



modal nominalism presents an underlying reality for mathematics that could be “sadly

uncooperative” (sensu Robust Realism).

5.3.4.1 Clear Obstacles to Affirming Methods?

I am aware of no convincing reasons for supposing that either of the three modal nominalist

strategies preclude the possibility of incorporating information which would help sort the

mathematically interesting theories and structures from the mathematically uninteresting

theories and structures. From Hellman’s and Field’s perspectives, there is nothing that

logically distinguishes one mathematical theory or structure from another. But that is not

to deny forthwith there are good mathematical reasons for opting to study one theory or

structure over another. The issue here is whether Hellman and Field are, respectively, com-

mitted to the claims that mathematics is just the study of structural possibilities, and that

mathematics is just an elaborate fiction. If so, then both views would appear to be method-

rejecting. But whether Hellman and Field actually make such claims is immaterial: There

is no reason why Hellman and Field could not take the relaxed attitude that recognizes a

role for considerations of mathematical fruitfulness. Indeed, by Maddy’s own admission,

that there are objective facts about what theories and concepts are mathematically deep

is consistent with treating these theories and concepts as logically on a par (Maddy 2011,

83). From Chihara’s perspective, there is nothing that metaphysically distinguishes one

true constructibility assertion from another. But again, that is not to deny forthwith that

Chihara is incapable of recognizing good mathematical reasons for opting to study one

kind of constructible theory or structure over another. It must be admitted that an account

of mathematical method will not fall trivially out of the metaphysics of either of these three

views, but the point here is that this does not itself preclude the possibility of theorizing

about mathematical method alongside modal nominalist theorizing about mathematical

metaphysics. No reason has been given for rejecting the possibility that modal nominalism,

like if-thenism, admits of “sophisticated” amendation. Thus, no reason has been given

for rejecting the idea that mathematical forms of justification could be used to distinguish



(and justify claims about) the mathematically salient modal claims.

5.3.4.2 Result- or Method-Rejecting?

Modal Structuralism presents the most promising candidate for a modal nominalist ac-

count of mathematics that avoids being both result- and method-rejecting. One reason for

this is that Modal Structuralism is method-affirming in a number of respects. For instance,

large tracts of algebra and topology focus almost exclusively on shared or structural prop-

erties of mathematical objects, rather than on the idiosyncratic features of, e.g., particular

groups, rings, or topological spaces. Moreover, Modal Structuralism accommodates math-

ematicians’ focus on abstraction, their absence of concern about foreseeable applications,

etc.19 A second reason is that the background assumption Modal Structuralism makes—

that mathematical theories and structures form subsets of the logical possibilities—is a

background assumption that is shared by Maddy’s depth-based account of mathematics.

Though fictionalism shares in the background assumption that mathematical theories

form subsets of the logical possibilities, nevertheless the fictionalist position is somewhat

riskier. As has already been discussed, there is a concern that indiscriminate reliance on

the fictionalist metaphor runs the risk of both method- and result-rejecting. For this reason,

fictionalism is the least promising candidate for a modal nominalist view that avoids result-

and method-rejecting.

Recall that Chihara’s Constructibility Theory does not attempt to describe the true

content of mathematical theories. That is, he does not argue that mathematical existence

assertions are, deep down, “really” constructibility claims. Instead, his motivation is to

. . . achieve for mathematics what philosophers of language hope to achievefor language: they seek to produce a coherent overall general account of thenature of mathematics (where by ‘mathematics’ I mean the actual mathematicspracticed and developed by current mathematicians)—one that is consistentnot only with our present-day theoretical and scientific views about the worldand also our place in the world as organisms with sense organs of the sortcharacterized by our best scientific theories, but also with what we know about

19Thanks to Geoffrey Hellman for suggesting these examples.



how our mastery of mathematics is acquired and tested. (2004, 6).

Importantly, he notes that such activity should not, “contradict any of our prevailing views

of science and scientific knowledge without very compelling reasons” (ibid., 2). If these

remarks are to be trusted, Chihara’s account is neither method- nor result-rejecting. How-

ever, though it can be granted that Chihara does not seek to overturn mathematical results

or undermine mathematical reasoning, it might be that the philosophical commitments of

Constructibility Theory place mathematically arbitrary limits on the size or complexity

of mathematical structures.20 For instance, (Jacquette 2004) argues that Constructibility

Theory cannot account for transinfinitary mathematics without relaxing its nominalistic

scruples. That Constructibility Theory fails to be method-rejecting would seem to depend

on the optimistic assumption that this view does not run into this kind of trouble.

It must be admitted that what has just been said is principally in defense of the three

modal nominalist theories of mathematics, rather than a defense of the motivations Chihara,

Field, and Hellman have used to develop these theories.21 Can anything be said in

defense of the claim that the motivation for modal nominalism is neither method- nor

result-rejecting? I am not certain if this is the best question to ask. As I have suggested

earlier, what nominalists tend to object to in the nominalism/platonism dispute are the

various philosophical presumptions platonists make about mathematics and mathematical

language, e.g., that mathematical assertions are true simpliciter, or that mathematical truths

presuppose or require the existence of mathematical objects. Thus, when the nominalist

(modal or otherwise) objects to Quine’s indispensability argument, or to Frege’s analysis of

mathematical language, she is not rejecting any of the methods or results of mathematics.

Rather, she is objecting to a certain kind of philosophical understanding of mathematics. She

might have one of several motivations for pursuing such objections: Some kind of a priori

prejudice against abstract objects, some set of epistemological concerns about knowledge

of abstracta, or the motivation to which I am most sympathetic: to account for the truths20Thanks to Susan Vineberg for bringing this issue to my attention.21Thanks to Michael McKinsey for suggesting this clarification.



of mathematics in an ontologically innocuous way. But the basic point here is that, once

it is granted (as Maddy seems happy to grant) that it is no presumption of mathematics

that mathematical objects exist, it is difficult to insist that the motivation for nominalism

(modal or otherwise) involves any kind of mathematical method- or result-rejecting.

Of course, if one takes seriously Maddy’s entreaty that mathematics be understood

solely on its own terms, then it is fair to ask why a naturalist of any stripe should be

willing to embrace any of the nominalist motivations just described, given that they

are principally philosophical motivations.22 But to pursue this line of reasoning further

would be to object to nominalism (including modal nominalism) on the grounds that

it fails to be method-contained, rather than on the grounds that it is either method- or

result-rejecting. The method-contained objection to modal nominalism will be discussed

in §4. However, I should think that it is largely immaterial whether the motivations

Chihara, Field, and Hellman have actually proposed are method- or result-rejecting. That

is because, even if these motivations are method- or result-rejecting, I believe that they

can be replaced in favor of a motivation that is clearly not method- or result-rejecting:

Namely, that modal nominalism facilitates the goal of providing an account of the content

of mathematical assertions that, just like mathematics, does not require that mathematical

objects exist. Of course, this goal is only worthy to the extent that modal nominalist

analyses of mathematical assertions do not raise the same epistemological difficulties that

the Robust Realist’s analyses raise. Whether this is so shall now be considered.

5.3.4.3 Underlying Reality is Sadly Uncooperative?

Does modal nominalism raise the possibility that the underlying reality of mathematics—

assertions about what is logically possible or what open-sentence tokens are constructible—

is “sadly uncooperative” in the sense that one could be perfectly mathematically justified in

asserting a mathematical claim, but nevertheless on that basis fail to be justified in making

22In full disclosure, this observation also applies to motivations for philosophical accounts of mathematicsmore generally.



the modal assertion that provides the metaphysical grounds for this claim?

Again, I believe that Modal Structuralism presents the best example of a modal nom-

inalist view for which this problem does not arise. The Modal Structuralist analysis of

mathematical assertions (as describing structural possibilities) might appear to be more

contentful than the analysis given by Maddy’s depth-based account, but I am not con-

vinced that this is actually the case. It must be remembered that Modal Structuralism is

committed to the primitive logical possibility of the existence of models of mathematical

theories. Thus the Modal Structuralist is committed to quite little in the way of a positive

characterization of mathematical structures. I am inclined to read Hellman as provid-

ing the most metaphysically austere conception of structure on record.23 Since Maddy

acknowledges that the facts of mathematical depth canvass theories and concepts that

constitute a subset of the logical possibilities (2011, 83), and further that classical logic and

logical consistency are important constraints on mathematical practice (alongside the facts

of mathematical depth) (ibid., 78), not even the Second Philosopher is immune from the

obligation to justify the same kind of possibility or consistency claims that underlie the

Modal Structuralist’s analysis of mathematics.

Now such justification will either accrue in the everyday practice of mathematics or it

will require some kind of extramathematical supplementation. If the justification for logical

possibility claims accrues in mathematics itself, then it is trivial that Modal Structuralism

avoids the possibility that the facts of logical possibility are sadly uncooperative. But if

the justification for logical possibility claims requires extramathematical supplementation,

then Modal Structuralism faces the possibility that the facts of logical possibility are indeed

sadly uncooperative—but then, Maddy (and seemingly everyone else) faces this possibility as well.

So either mathematics itself provides sufficient reason for thinking that various theories

and structures are logically possible, or else it is not possible to provide a full account

of the constraints on mathematical practice by appealing solely to the methodology of

23Cf. (Hellman 2005) and (McLarty 2008a, 2008b).



mathematics. Modal structuralism, therefore, is no more “uncooperative” than Maddy’s

depth-based metaphysics.24

Field’s fictionalism—at least its account of pure mathematics—can be given a similar

defense to the extent that it interprets mathematical knowledge as a genus of logical

knowledge. This knowledge comes in two forms: Knowledge of the consistency of

mathematical theories, and knowledge about which theorems are logical implications

of mathematical axioms. For Field, both consistency and implication are themselves

understood to be primitive modal notions. Importantly, he does not take himself to be

replacing the ordinary notions of consistency and implication with new, primitive modal

notions. Rather, he argues that the ordinary notions of consistency and implication are

already primitive modal notions. If that argument is correct, then Field, like Hellman and

Maddy, can take advantage of internal mathematical justifications for assertions about

what is logically consistent, if any such justifications are available.

Chihara’s Constructibility Theory poses the greatest risk of positing a metaphysics that

is “sadly uncooperative.” In a sense, this is a pseudo-problem, because Constructibility

Theory is not offered as a genuine account of the underlying reality of mathematics,

but instead as an account of how it is possible to engage in mathematical reasoning

without quantifying over mathematical objects. Nevertheless it can still be objected

that the metaphysical modalities invoked by Chihara are uncooperative in the sense that

Chihara cannot guarantee that it is metaphysically possible to construct a sufficient number

and variety of open-sentence tokens to capture constructibility correlates of all accepted

24Though it is correct to say that Maddy and Hellman should each be concerned to justify beliefs about theconsistency of mathematical theories, it would not be correct to say that they both accept the same analysisof the logical and/or modal notions. Maddy adopts a rather unusual position on the status of classical logic(2007, 197-302). She believes that the logical truths of a system resembling Kleene three-valued logic canbe justified through appeal to facts about the structure of the physical world and facts about innate humancognitive abilities. Classical logic is justified only when certain idealizing assumptions have been made(one being a prohibition against vague predicates); such idealizing assumptions are made because, “theymake it possible to achieve results that would otherwise be impossible or impractical” (ibid., 288). Thus, onMaddy’s view, the ground of classical logic is ultimately pragmatic. Though I have very great reservationsabout this account of classical logic and the extent to which it avoids the epistemological problems raised byprimitivism about logic and modal logic, I do not have space to describe them here.


Reflections on Maddy 265 The Method-Affirming Objection: Coda

mathematical results. It seems rather unlikely that the methods of mathematics have much

to say about constructing open-sentence tokens, and it is therefore extremely doubtful

that one could appeal to mathematics in order to justify the constructibility assertions that

underlie Constructibility Theory.

I think the lesson here is that neither Maddy nor anyone else can preclude the possibility

that modal facts in general are uncooperative.25 So although the task before Chihara—

justifying constructibility assertions—appears to be different in kind from the task before

Hellman, Field, and Maddy—justifying logical possibility assertions, it is far from clear

that these tasks are distinguishable on the basis of their level of difficulty. Indeed, one of

the morals of chapter three is that both kinds of modal assertions raise a very similar set of

metaphysical and epistemological problems. In any case, it can at least be maintained that

Modal Structuralism and fictionalism present a metaphysics that is no more uncooperative

than Maddy’s depth-based account, which means that it would inappropriate to object

to these views on the grounds that there is a great gulf between the modal facts and the

accepted assertions of mathematics.

5.3.5 The Method-Affirming Objection: Coda

That modal nominalism fails to be method-affirming, then, does not constitute a compelling

reason for thinking that modal nominalism is either method- or result-rejecting. (Though

I acknowledge that this claim is strongest in the case of Modal Structuralism, since it is

the only account that does not clearly face any of the three kinds of objections discussed

above.) It is open to a modal nominalist to theorize about mathematical methodology

alongside her metaphysical account of the nature of mathematical objects, and there is no

prima facie reason to believe that these dual pursuits will conflict with one another. Just as

in the case of if-thenism, there is nothing inherently problematic about the fact that modal

nominalism fails to be method-affirming.

Given, then, that modal nominalism, in at least some of its formulations, is neither25This is a point that I will revisit in my response to the method-contained objection to modal nominalism.


Reflections on Maddy 266 The Method-Contained Objection

method- nor result-rejecting, might it be further argued that modal nominalism is con-

sistent with Second Philosophy? It must be admitted that, as far as compatibility with

mathematical practice is concerned, being method-affirming functions as a mere prophy-

lactic against method- and result-rejecting. Does a similar judgment hold for consistency

with Second Philosophy? That a “sophisticated” if-thenism appears to be compatible

with Second Philosophy (ibid., 99) suggests, by analogy, that modal nominalism is also

compatible with Second Philosophy.

However, that modal nominalism is compatible with Second Philosophy is difficult to

square with Maddy’s use the method-contained component of her entreaty that mathe-

matics be understood on its own terms. To see why this is so it is necessary to examine

the two views Maddy takes to be compatible with Second Philosophy—Thin Realism and

Arealism—and it is also necessary to examine Maddy’s reasons for thinking that there is

no substantive difference between these views. I shall argue that, as far as compatibility

with mathematical practice is concerned, being method-contained is a mere prophylactic

against method- and result-rejecting. Nevertheless the method-contained component fac-

tors prominently in Maddy’s assessment that, all things considered, there is no substantive

difference between Thin Realism and Arealism. Therefore modal nominalism, because it

fails to be method-contained, is not compatible with Second Philosophy.

5.4 The Method-Contained Objection to Modal Nominalism

Recall that the method-contained objection to modal nominalism states that modal nomi-

nalism is objectionable because modal nominalist accounts of mathematics are not endemic

to mathematical methodology but instead require explicitly philosophical resources and

motivation. Are views that fail to be method-contained objectionable for this very reason?

And are they incompatible with Second Philosophy for this very reason? Or is failing to

be method-contained, like failing to be method-affirming, just a defeasible indicator of

method- and result-rejecting?


Reflections on Maddy 267 The Method-Contained Objection

If the function of the method-contained component of Maddy’s entreaty that mathe-

matics be understood on its own terms is, like the method-affirming component, just a

prophylactic against method- and result-rejecting, then the defense of modal nominalism

from the previous section can be appropriated. The method-contained component is

likely to combine the method-/result-rejecting and “sadly uncooperative” concerns: To go

beyond mathematics creates the risk that one’s metaphysics is sadly uncooperative, threat-

ening to discount the idea that internal mathematical forms of reasoning provide sufficient

justifications for mathematical assertions. It should be clear that modal nominalism, at

least in certain of its formulations, carries minimal risk on both counts.

But, as it turns out, the function of the method-contained component of Second Philos-

ophy is altogether distinct from the method-affirming component, and for this reason I

think that it is worthwhile to examine how the method-contained component factors into

Maddy’s discussion of the two positions she takes to be inherently compatible both with

mathematics and with Second Philosophy—Thin Realism and Arealism. These two views

share a common metaphysical core, viz., that under both views the facts of mathematical

depth comprise the underlying reality of mathematics. In this regard Thin Realism and

Arealism are both method-affirming. Maddy uses the method-contained component to

argue that there is no substantive difference between these views, because what distin-

guishes them from one another are the claims they make about mathematics from a more

general empirical perspective (i.e., from outside mathematics): The Thin Realist claims

that mathematics is a body of truths, the Arealist denies that mathematics is a body of

truths. Maddy officially endorses only the common methodological core of Thin Realism

and Arealism. In the final two sections I make a provisional case that this use of the

method-contained component unveils Second Philosophy as an implausibly strong, and

possibly incoherent form of naturalism, and that modal nominalism is compatible with

forms of naturalism which omit the method-contained component of Maddy’s entreaty.


Reflections on Maddy 268 Thin Realism

5.4.1 Thin Realism

Thin Realism is a moderate platonism that has many points of contact with Burgess’s

moderate platonism, discussed in the previous chapter. Maddy’s motivation for devel-

oping a “thin” version of realism is to produce, “a satisfying form of realism without the

shortcomings of the Robust versions” (ibid., 60), where such satisfaction accrues when

this realism, “genuinely accounts for the nature of set-theoretic language and practice,”

and, “respects the actual structure of set-theoretic justifications” (ibid., 61). As discussed

above, the shortcomings of robust versions of realism consist in their various forms of

method- and result-rejecting. If the overriding virtue of Thin Realism is that it lacks these

shortcomings, then it would appear that the primary force recommending Thin Realism is

that the view avoids method- and result-rejecting.

The main theses of Thin Realism are that set theory expresses a body of truths; that

sets exist; and that, “sets are just the sort of thing set theory describes; this is all there

is to them; for questions about sets, set theory is the only relevant authority” (ibid., 61).

Moreover, on this view, “set-theoretic methods are the reliable avenue to the facts about

sets. . . no external guarantee is necessary or possible” (ibid., 63), and further that, “the

evidence for the existence of sets is all and only linked to their mathematical virtues, to

the mathematical jobs they are able to perform” (ibid., 73). These descriptions of Thin

Realism rather directly identify the view as both method-affirming and method-contained.

However, I think that the issue of whether Thin Realism is method-contained is rather

more subtle than the above quotations indicate.

The Thin Realist’s reason for regarding mathematics as a body of truths is not because of

any reasons internal to the practice of mathematics, but instead, “because of [mathematics’]

particular relations with the defining empirical inquiry from which she begins” (ibid., 70).

The idea here is that the Second Philosopher, with a starting point in natural science, bears

witness to the remarkable interplay between science and mathematics, eventually coming

to respect mathematics as a successful, freestanding discipline, itself worthy of the honorific



‘scientific.’ A decision is then made to extend the applications of concepts like ‘truth’ and

‘existence’ to the assertions of set theory and the rest of mathematics. Thus, the decision

to regard mathematics as a body of truths arises from the broader perspective afforded

by the Thin Realist’s empirical origins. It must be admitted, then, that Thin Realism is not

method-contained in a purely mathematical sense—but it is ostensibly method-contained

in a broader empirical sense, because the reasons for thinking that mathematics is a body

of truths are alleged to come from the broader empirical perspective.

Concerning the nature of sets in particular, they are, “maximally effective trackers of

certain strains of mathematical depth” (ibid., 82). On this account of set theory, depth

functions as a guide to existence:

. . . the fact of measurable cardinals being mathematically fruitful in ways x,y,z(and these advantages not being outweighed by accompanying disadvantages)is evidence for their existence. Why? Because of what sets are: repositoriesof mathematical depth. They mark off a mathematically rich vein within theindiscriminate network of logical possibilities. (ibid., 82-3)

Thus, evidence for the depth or fruitfulness of an axiom is also evidence for the truth of

that axiom and thereby is also evidence for the existence of those objects whose existence

is implied by that axiom. Thin Realism is thus a combination of realism about the facts

of mathematical depth and realism about mathematical objects, under which knowledge

of the former serves as evidence for knowledge about the latter. The claim here is that all

deep theories and concepts have exemplifications in the Thin Realist’s ontology. Linking

existence claims to the facts of mathematical depth in this way allows the Thin Realist

to avoid the epistemological problems raised by the Robust Realist’s desire for a further,

extramathematical justification for these existence claims. However, one might wonder if

the converse is true—whether all true mathematical existence claims are included in some

or other deep mathematical theory.26 For example, According to Maddy, the ground of

elementary arithmetic comes not from the facts of mathematical depth but instead from

26Thanks to Susan Vineberg for raising this issue.



more mundane facts about the physical world and human cognition (2007, 318-328). If

Maddy’s Thin Realist also adopts realism about the objects referred to in the existence

claims of elementary arithmetic, then it would appear as though depth is not the only

guide to mathematical existence.27

For this reason care must be taken to distinguish questions about the existence or not

of mathematical objects in general and the truth or not of mathematics in general from

questions about the existence or not of specific mathematical objects and the truth or not

of specific mathematical assertions. The role of mathematics in science is what validates

mathematics as a body of truths (along with the existential ones); the facts of depth serve

as evidence only in specific cases. The possibility remains, then, that there is more to the

Thin Realist’s ontology than can be gleaned from the facts of mathematical depth, which

would further support my contention that Thin Realism is not method-contained in a

purely mathematical sense. This would be especially troubling if it turned out that the

Thin Realist’s trivialist mathematical epistemology cannot access the full domain of the

Thin Realist’s ontology. To solve the problem for existence claims about sets is not to solve

(or give the form of the solution for) the problem for all mathematical existence claims—

unless one assumes in addition that only mathematically deep theories and concepts have

exemplifications in the Thin Realist’s mathematical ontology. Since it is unclear whether

Maddy qua Thin Realist makes this assumption, I will not pursue this concern further.

Why is Thin Realism open to the Second Philosopher? Largely because of the mini-

mality of its metaphysical and epistemological commitments. Maddy identifies it as the

“simplest” realistic hypothesis about the nature of sets (Maddy 2011, 61). Given that sets

track deep strains of mathematics, and that good mathematical reasoning (usually) ac-

cesses these strains, it becomes something of a triviality that good mathematical reasoning

27Or, at least, depth is not the only guide to existence claims about numbers. If the facts of mathematicaldepth are what distinguish mathematical truths from non-mathematical truths, and if elementary arithmeticis not mathematically deep, then the numbers of elementary arithmetic are not, strictly speaking, mathematicalobjects. Of course, elementary arithmetic is embedded in numerous other theories that Maddy woulddescribe as deep, so perhaps my distinguishing between number-existence and mathematical-existencepoints to nothing of significance. Cf. (ibid., 362).



arrives at justified beliefs about assertions sets. To this extent, at least, Thin Realism is

method-contained. But that these are inherently virtuous qualities of Thin Realism con-

flicts with Maddy’s explanation for why one should avoid appealing to extramathematical

considerations:

Philosophers often begin from a more elevated perspective: rather than ex-amining the day-to-day practices, they content themselves with classifyingmathematics as a non-empirical, a priori discipline, concerning a robust ab-stract ontology, then begin to wonder how we could possibly come to knowsuch things, how what mathematicians actually do could have any connectionto the subject matter they’re attempting to describe. . . Roughly put, they beginwith the metaphysics and are led to confusion about the methods. In contrast,the Second Philosopher begins with the methods, finds them good, then devisesa minimal metaphysics to suit the case. (ibid., 86)

What is suggested here is that, in being method-contained, Thin Realism removes most (if

not all) of the philosophical interference present in competing views, e.g., in Robust Realism.

The problem with beginning from an “elevated perspective” is precisely that starting

from this perspective dramatically increases the chances one will endorse method- and

result-rejecting (because, e.g., mathematical methods might conflict with first-philosophical

intuitions about what mathematical knowledge must be). So a virtue of views that are

method-contained is that they dramatically reduce or eliminate entirely the risk of method-

and result-rejecting. But this would be to laud being method-contained for its prophy-

lactic qualities, rather than as an inherently virtuous quality of philosophical accounts of

mathematics.

Another example of a use of the method-contained component is when Maddy claims

that the justification for claims about mathematical metaphysics must arise from mathemat-

ical methods—and not the reverse (ibid., 87)—but this assertion is immediately followed

by a reminder that purely metaphysical considerations have not restricted the “free pur-

suit of pure mathematics” (ibid.). Nowhere in her discussion of Thin Realism does the

method-contained component play an independent role in establishing the virtues of Thin

Realism—Thin Realism is always virtuous by contrast with Robust Realism, or by contrast


Reflections on Maddy 272 Arealism

with some other view that is objectionable from the Second-Philosophical point of view

for being in some kind of conflict with mathematics.

5.4.2 Arealism

In contrast with Thin Realism, Maddy argues that it also consistent with mathematical

practice and with Second Philosophy to say that mathematics is not a body of truths. Her

label for this alternative position is Arealism. The Arealist and the Thin Realist both agree

about what makes for sound mathematical methodology. The Arealist does not contest

that the pressures guiding mathematical concept formation and axiom selection are the

facts of mathematical depth—the Arealist, like the Thin Realist, is a realist about the facts

of mathematical depth. Thus Arealism, like Thin Realism, is method-affirming. Further,

the Arealist does not approach mathematics with a prior prejudice against abstract objects,

unlike Ockhamite forms of nominalism (ibid., 97), which suggests that Arealism is also

method-contained.

But just like in the case of Thin Realism, it is not clear that Arealism is method-contained

in a purely mathematical sense. This is because the Arealist denies that sets exist (ibid.,

100). This judgment is based, not on any explicitly mathematical grounds, but instead on

the attitude the Arealist adopts toward mathematics from her initial empirical perspective:

. . . [the Arealist] begins her investigation with ordinary perception, graduatesto more sophisticated forms of observation, theory formation. . . eventually sheturns to mathematical methods, and from there, to the pursuit of mathematicsitself. . . she finds her mathematical inquiries broadening to include structuresand methods without immediate application, which eventually leads her to settheory along the path of Cantor, Dedekind, Zermelo, and the rest. . . When shenotices that its methods are quite different, that its claims aren’t supported byher familiar observation, experimentation, theory formation, and so on. . . mightshe not simply conclude that whatever its merits, pure mathematics isn’t in thebusiness of uncovering truths? (ibid., 88-9)

In other words, the Arealist bears witness to the remarkable methodological differences

between natural science and mathematics, differences that suggest to her that it is inap-

propriate to apply words like ‘true’ and ‘exists’ to mathematics and mathematical objects;


Reflections on Maddy 273 Arealism

Maddy has described the view as, “taking back in one’s scientific moments what one

says in one’s mathematical moments” (2005a, 368; 2007, 390). Arealism, therefore, is not

method-contained in a purely mathematical sense, but, as in the case of Thin Realism, it is

method-contained in a broader empirical sense.

Though not identical to traditional forms of nominalism, Arealism does face a similar

set of hurdles in accounting for scientific applications of mathematics. Maddy’s Arealist

account of applications resembles other fictionalist accounts of applications.28 The basic

idea is that, “what we need to know isn’t so much that the advanced mathematics is true,

but that the more esoteric features it reveals will continue to be effective in modeling

the world” (Maddy 2011, 92). I do not intend to discuss the problem of application here.

However, it is important to note that this kind of fictionalist account of modeling is not

clearly unproblematic,29 and if the Arealist’s account of applications ultimately proves

untenable, then Arealism does not appear to be a viable option for the Second Philosopher.

Nevertheless, insofar as, “the truth (or not) of mathematics is irrelevant to explaining its

role in scientific application, it appears that Arealism is open to our Second Philosopher”

(Maddy 2011, 96).

The permissibility of Arealism derives largely from the fact that it leaves the practice

of mathematics intact—as a realist about the facts of mathematical depth, the Arealist is

fully capable of coming to sound judgments about which axioms and concepts make for

deep or fruitful mathematics. One might question whether this is so. Is it not part of the

practice of mathematics to regard its accepted assertions as true (and its refuted claims as

false)? Would it not be a form of method- or result-rejecting to deny this? Just what is the

Arealist denying when she denies that mathematics is a body of truths? Note that Maddy

describes terms like ‘true’ and ‘exists’ as honorifics (ibid., 111; 2005a, 368); seemingly to

be bestowed (or not) at the whim of the attitude the Second Philosopher adopts from her

empirical perspective.

28See, e.g., (Balaguer 1998) and (Leng 2010).29See (Vineberg 2008).


Reflections on Maddy 274 There is no Difference Here

Whether this is a problem for Arealism depends on whether there is evidence that a

plausible use of mathematical methodology is to assert the truth simpliciter of mathematical

existence claims. At least on Maddy’s perspective, the answer to this question seems to

be no—even under Thin Realism. When the Thin Realist states that mathematics is a

body of truths, she does not do so for internal mathematical reasons. Instead this is the

product of her judgment about the relationship between mathematics and science. If there

is an internal mathematical conception of truth (or some other honorific that functions like

truth), then she happily adopts it as continuous with whatever conception of truth she has

inherited from her empirical origins. Similarly, when the Arealist denies that mathematics

is a body of truths, she does not do so for internal mathematical reasons. If there is an

internal mathematical conception of truth, she will not deny the use of this conception

as it applies to claims within mathematics. But since, “her well-developed methods of

confirming existence and truth aren’t even in play” in mathematics (Maddy 2011, 89), she

decides to regard this internal mathematical conception as discontinuous with her ordinary

empirical conception of truth. If this practice turns out to be method- or result-rejecting,

then so much the worse for Arealism. Otherwise, the view is only involved in metaphysics

to the extent required by its inherent realism about the methodologically-driven facts

of mathematical depth; and the benefit of this metaphysical austerity is that it prevents

method- and result-rejecting.

5.4.3 There is no Difference Here

One of the more fascinating allegations of (Maddy 2011) is that there is no substantive

difference between Thin Realism and Arealism—both views are “indistinguishable at the

level of method” (ibid., 100), and are ultimately indistinguishable to Maddy’s naturalist.

Maddy’s reasoning here is rather subtle, and relies on claims she makes from both a per-

spective within mathematics as well as a perspective of an inquirer assessing mathematics

from a more general empirical or scientific setting. To borrow the eponymous phrase of

(Ferreiros 2010), this parity result arises from the adoption of naturalism both in and about



mathematics. The following passage is paradigmatic of Maddy’s observations from within

mathematics:

This methodological agreement [between Thin Realism and Arealism] reflects adeeper metaphysical bond: the objective facts that underlie these two positionsare exactly the same, namely, the topography of mathematical depth. . . For bothpositions, the development of set theory responds to an objective reality—andindeed the same objective reality. (Maddy 2011, 100)

Thus, any decision between these two views is not to be arbitrated on the basis of the

methodology of mathematics, for it is this methodology that is singularly incapable of

recognizing any relevant distinction between Thin Realism and Arealism.

At this point Maddy has established her parity result from within mathematics. But,

as it has already been mentioned, Maddy also recognizes the legitimacy of a perspective

outside of mathematics, viz., the empirical origins of the Second Philosopher. The dispute

between the Thin Realist and the Arealist, “takes place not within set theory, but in the

judgments they form as they regard set-theoretic language and practice from an empirical

perspective” (ibid., 100). From this perspective, Maddy claims, there is no decisive reason

to favor Thin Realism over Arealism, or vice versa—nothing about the role of mathematics

in the scientific world view forces the Second Philosopher to apply the honorifics ‘true’

and ‘exists’ to the assertions of mathematics. This is because

. . . the two idioms are equally well-supported by precisely the same objectivereality: those facts of mathematical depth. These facts are what matter, whatmake pure mathematics the distinctive discipline that it is. . . Thin Realism andArealism are equally accurate Second-Philosophical descriptions of the natureof pure mathematics. They are alternate ways of expressing the very sameaccount of the objective facts that underlie mathematical practice. (ibid., 112)

Not even outside of mathematics does there arise a legitimate preference for Thin Real-

ism over Arealism (or vice versa). So although both views are method-contained in a

broader empirical sense, the broader empirical methods parlay agnosticism. Since both the

methodology of mathematics and the empirical perspective prove facile, there is nothing

further for Maddy’s naturalist to say on matters. Of course, if a salient difference were



discovered between Thin Realism and Arealism, then the Second Philosopher would have

a reason to choose between the views. Thus Maddy should not be read as claiming that

it is impossible in principle, from the empirical perspective, to decide between alternate

metaphysical accounts of mathematics. Rather, if Maddy is right, it simply happens to be

the case that there are no salient differences between Thin Realism and Arealism.

I would like to offer the following analysis of Maddy’s parity result: Maddy’s claim

that the facts of mathematical depth are the only facts that “matter” factors prominently

in reaching her conclusion that, both from within mathematics and from without, there

is no substantive difference between the two views. Maddy thinks that it is a crucial

finding that none of the methods of mathematics proper decide between Thin Realism and

Arealism. This is evidence that, for the Second Philosopher, the fact that Thin Realism

and Arealism are method-contained in a broader empirical sense is ultimately of little

significance, because what distinguishes these two views cannot be appreciated from a

perspective that is method-contained in a strictly mathematical sense. The only aspects

of Thin Realism and Arealism that are ultimately positively sanctioned by the Second

Philosopher are the strictly mathematical components of these views—that the facts of

mathematical depth comprise the underlying reality of mathematics, that it is part of the

methods of mathematics to reveal these facts, etc.

Both Thin Realism and Arealism, then, constitute examples of the idea that failing

to be method-contained in a strictly mathematical sense does not induce method- and

result-rejecting (malevolently or otherwise). But the Second Philosopher still seeks to

excise the extramathematical components of Thin Realism and Arealism, even though

these components pose no risk of method- and result-rejecting. For Maddy, then, the

claim that the facts of mathematical depth are the facts that matter—the method-contained

component of her entreaty to understand mathematics on its own terms—plays more than

a merely prophylactic role. Excised of this component of her entreaty, Maddy’s naturalism

cannot establish the indistinguishability of Thin Realism and Arealism.


Reflections on Maddy 277 The Method-Contained Objection: Coda

But why suppose that the facts of mathematical depth are the facts that matter? Why

does the Second Philosopher insist that the method-contained component plays an inde-

pendent and decisive (or, perhaps, indecisive?) role in apportioning merit to accounts of

mathematics? Without a justification for the claim that mathematical metaphysics is wholly

constituted by the pressures underlying mathematical concept formation and axiom selec-

tion, it simply does not follow that there is no substantive difference between Thin Realism

and Arealism. I am forced to conclude, then, that Maddy intends the method-contained

component of her entreaty to stand on its own, as an independent and vital feature of

Second Philosophy. It follows that, since modal nominalism fails to be method-contained, then it

is not consistent with Second Philosophy, even though modal nominalism does not (in certain of its

formulations) appear to be either method- or result-rejecting.

5.4.4 The Method-Contained Objection: Coda

I will not contest the claim that modal nominalism, because it fails to be method-contained,

is inconsistent with Second Philosophy. Nevertheless there are four reasons why I maintain

that modal nominalists should not be bothered by this inconsistency.

First, modal nominalism appears to be defensible by analogy with Thin Realism and

Arealism on the grounds that its failure to be method-contained is not sufficient evidence

for thinking that modal nominalism is either method- or result-rejecting. There is a

potentially crucial difference between modal nominalism and Maddy’s favored views,

however. And this is that both Thin Realism and Arealism are, allegedly, method-contained

in a broader empirical sense, whereas no such claim has been made on behalf of modal

nominalism. But even if modal nominalism fails to be method-contained in a broader

empirical sense—and whether this is so is very much an open question—I would still

insist, in light of the results of §3, that modal nominalism poses very little risk of method-

and result-rejecting.

Second, there is reason to suppose that the method-contained component of Maddy’s

entreaty to understand mathematics on its own terms is not a coherent feature of Second


Reflections on Maddy 278 The Method-Contained Objection: Coda

Philosophy. It is worth restating that Maddy herself recognizes that a full account of the

objective constraints on mathematical practice involves legitimizing not only the facts of

mathematical depth but also legitimizing claims of logical consistency (Maddy 2011, 73; 78).

The latter of which is, presumably, the only significant constraint on the Modal Structuralist

version of modal nominalism. So it is not entirely clear that modal nominalism (at least via

Hellman) requires more than the Second Philosopher’s depth-based metaphysics requires,

at least from a logical or conceptual point of view. If the justification for logical consistency

claims is to come from somewhere other than their pragmatic uses in mathematics, then

everyone must look beyond mathematics to fully understand the nature of mathematics—

the facts of depth included.30 If this poses any kind of risk of method- and result-rejecting,

then it is a risk that Maddy shares with the Modal Structuralist.

Third, I suspect that Maddy’s entreaty that philosophical accounts of mathematics be

contained by the methods of mathematics unveils Second Philosophy as an implausibly

strong form of naturalism. For reasons already given, I doubt that it is possible to produce

a coherent philosophy of mathematics using only the methods of mathematics.

Fourth, and finally, given that modal nominalism appears to be compatible with every

element of Second Philosophy except the method-contained component of the impulse to

understand mathematics on its own terms, it stands to reason that modal nominalism is

consistent with any form of naturalism that adopts only the weaker Second Philosophical

principles—to avoid method- and result-rejecting, and to be method-affirming—whether

individually or jointly.

Reasons three and four raise deep questions about the overall coherence of Maddy’s

naturalism and more generally about which components of Maddy’s naturalism represent

desirable features of the naturalistic perspective. In the short space remaining I cannot

hope to provide any decisive objections to Second Philosophy, nor can I hope to provide a

detailed circumscription of the alternative naturalist perspectives that are available, but I

30Of course, if the legitimization of claims of logical consistency is part of mathematical method, then therewould be grounds for thinking that Modal Structuralism is method-contained.


Reflections on Maddy 279 Too Much Naturalism?

can at least relay some provisional remarks on both items.

5.5 How Much Naturalism is Too Much Naturalism?

One thing I would like to suggest is that it is not be possible to produce a coherent philoso-

phy of mathematics excised of everything except the methods of mathematics. Something

extramathematical appears required for producing an account of logical consistency. Fur-

ther evidence that the facts of mathematical depth do not provide a self-sustaining basis for

the metaphysics of mathematics was alluded to during my discussion of Thin Realism, viz.,

that there are both deep and shallow truths of mathematics.31 Are basic truths of arithmetic

and analysis any less mathematical for not generating much in the way of mathematical

interest? I should think not—but then it is not clear that the facts of mathematical depth

are reliable guides for distinguishing between mathematical and non-mathematical claims.

If that is right, then there is no guarantee that Maddy’s method-contained, depth-style

trivialist epistemology will suffice for all mathematical claims—it would seem only ap-

propriate for deep mathematical claims. For this reason it appears unduly restrictive to

object to a metaphysical account of mathematics simply because it outstrips the province

of mathematical methodology, i.e., simply because it fails to be method-contained.

But acknowledging this would place Second Philosophy on very shaky foundations—is

it not a central tenet of Second Philosophy that the only methods for understanding and

evaluating mathematics are mathematical methods? Would challenging this conviction

not simply be tantamount to objecting to Second Philosophy (and to naturalism more

generally)? Perhaps. But if the Second Philosopher does not have a cogent case that

all extramathematical analyses of mathematics are assailed by irrelevance and conflict,

then merely highlighting a view’s appeal to something extramathematical cannot be a

decisive reason—even for the Second Philosopher—to object to a philosophical account of

mathematics. Maddy has provided compelling reasons for rejecting certain extramathe-

matical perspectives, e.g., Quine’s views on ontological economy in set theory, but a general

31Thanks again to Susan Vineberg for this observation.



prohibition has only been established by fiat.

One claim I would like to make is that naturalism should not simply be the dogmatic

rejection of first philosophy or of extramathematical considerations. The rejection of such

considerations should be decided on a case-by-case basis. For instance, by summarily

rejecting first-philosophy, Maddy does evict numerous considerations that are irrelevant to

mathematical practice (e.g., preoccupation with establishing the truth of mathematics on a

priori grounds), but she also evicts potentially helpful first-philosophical considerations.32

One might even go so far as to argue, contra Maddy, that philosophical beliefs are part of

the evidential structure of the subject. Favorable cases do suggest themselves—for instance,

Godel’s belief in a “well-determined reality” (1983, 476) underlying the set theoretic axioms

(which was inseparable from his interest in whether the Continuum Hypothesis was true

or false of this reality), or the beliefs of Heyting and Brouwer, which led to the development

of intuitionistic logic and mathematics.33 Maddy is of course skeptical on this point:

Given the wide range of views mathematicians tend to hold on these matters, itseems unlikely that the many analysts, algebraists, and set theorists ultimatelyled to embrace sets would all agree on any single conception of the nature ofmathematical objects in general, or of sets in particular; the Second Philosopherconcludes that such remarks should be treated as colorful asides or heuristicaides, but not as part of the evidential structure of the subject. (Maddy 2011,52-3)34

Maddy is right to point out that mathematicians hold a “wide range of views” on matters,

but the mere fact of philosophical disunity in the mathematical community is not alone

evidence that philosophical (including first-philosophical) considerations are always or

are even usually irrelevant to mathematics and its methodology. It might just be that

some mathematicians are right and others are wrong, when it comes to their philosophical

beliefs; and it might be that philosophical beliefs can be mathematically productive, even

if they are wrong.35

32Cf. (Moore 2006).33Thanks to Susan Vineberg for suggesting these cases.34See also (Maddy 1997, 192-3).35This latter point echoes the observation of (Schlimm 2013) that, despite the value of consistency to the



Why, then, does Maddy object so strongly to the first-philosophical point of view? It

cannot simply be that the perspective is first-philosophical! My sole aim in distinguishing

between the ideas that mathematics should be understood and evaluated on its own terms

(in concert with the decomposition of the former into the method-affirming and method-

contained components, and the decomposition of the latter into the method- and result-

rejecting components) was to provide a rationalization for Maddy’s rejection of various

philosophical accounts of mathematics. But the various philosophical accounts that utilized

extramathematical resources were seen to cause problems with practice only when they were either

method- or result-rejecting. Failing to be either method-affirming or method-contained is not

essentially linked to being method- and result-rejecting. Why, then, is the most reasonable

perspective one that abandons first philosophy entirely? Why is it not more reasonable

to apply something like a “no-harm” policy regarding first philosophy? It is true that

some uses of first philosophy malevolently conflict with mathematics, and the Second

Philosopher (along with all dutiful naturalists) should censure such uses; but that is not

the situation that appears to be presented by Hellman’s Modal Structuralism (and perhaps

also Chihara’s Constructibility Theory). I have difficulty understanding what the objection

to these views is, if it is something other than anti-first-philosophical foot stamping.

What, then, is the ground of the method-contained component of Maddy’s entreaty to

understand mathematics on its own terms? It does not appear, for instance, to be a claim of

mathematics proper that the metaphysics of mathematics is wholly constituted by the facts

of mathematical depth. Rather, this is what (Daly and Liggins 2011) would describe as a

“philosophical imposition” concerning the nature of mathematical objectivity—albeit one

that happens to avoid many of the (alleged) irrelevancies of Maddy’s ontology-preoccupied

predecessors. If it is indeed true that the only resources available to the Second Philosopher

a propos of mathematics are the methods of mathematics, then since the method-contained

mathematical community, inconsistent sets of axioms may provide the starting point for fruitful mathematicalwork. Cf. (Easwaran 2008), where it is argued that mathematicians adopt axioms to avoid philosophicaldisputes.



component is not endemic to mathematics, the method-contained component is self-

undermining. If Maddy is genuinely committed to the method-contained component of

her entreaty, then her naturalism is implausibly strong.

But perhaps the Second Philosopher intends to leave just enough philosophical wiggle

room to advance or otherwise justify the method-contained component of Maddy’s en-

treaty. This, however, only serves to raise the question as to why the Second Philosopher

is permitted to make this philosophical claim, but is not permitted to make the kinds of

philosophical claims modal nominalists make. I do not know of any clear way of address-

ing this question, aside from either (a) reiterating that accounts of mathematics ought

to be method-contained, or (b) reanimating the worry that when an account fails to be

method-contained that it thereby induces method- and result-rejecting. Both strategies are

unconvincing. Are there other options available? One might propose that considerations

of simplicity offer support for the method-contained component and its sequester on

extramathematical metaphysical theorizing about mathematics. Since a depth-based meta-

physics accounts for the metaphysics of mathematics and the facts of mathematical depth

in one fell swoop, it appears to qualify as a simpler account of mathematics when compared

to modal nominalism. But why, in the circumstances at hand, should simplicity count as

a virtue? And what notion of simplicity should be invoked? One argument for thinking

that simplicity is a virtue is the observation that, as the complexity of one’s mathematical

metaphysics increases, so too does the likelihood that one’s metaphysics will run into

conflict with mathematical practice. But this is just to say that simplicity—and ultimately,

being method-contained—is a mere prophylactic against method- and result-rejecting. So

it is difficult to see at the same time both what the justification for the method-contained

component is, and why it should be thought that violating the method-contained compo-

nent is inherently problematic. Though the method-contained component may be advanced

as a valuable component of Second Philosophy, I do not see that there is any independent

evidence for this attribution of value. It would be special-pleading to insist that the only


Reflections on Maddy 283 Conclusion

philosophical claim one is allowed to make about mathematics is that an understanding of

the subject must be confined by its own methods.

It would appear then, that justifying the method-contained component of Maddy’s

entreaty presents serious difficulties. Not only does it seem implausible that one could

produce a coherent metaphysical account of mathematics using only the methods of

mathematics, it is also implausible that one could motivate, using only the methods

of mathematics, the idea that one should use only the methods of mathematics when

constructing a metaphysical theory of the subject. What this suggests is that any form

of naturalism that embraces the method-contained component of Maddy’s naturalism

is implausibly strong—the method-contained component of Maddy’s entreaty must be

dropped. If Maddy’s naturalism is implausibly strong, then what might weaker forms of

naturalism look like? And with what kinds of naturalism would modal nominalism be

compatible?

5.6 Conclusion: Naturalism and Modal Nominalism

I should think that Maddy’s entreaty that mathematics be evaluated on its own terms

comprises the foundation of basic naturalist dogma. A naturalism that permitted wanton

philosophical upheaval of mathematical methods or results would be no naturalism at all.

Thus, the weakest form of naturalism seems to be one which merely prohibits method-

and result-rejecting, at least in their malevolent instances. I hope I have made a plausible

case for thinking that modal nominalism poses little risk of method- and result-rejecting,

and would therefore be compatible with this weak form of naturalism.

Nevertheless I suspect that Maddy is correct to judge that an account of the actual

methods of mathematics is a philosophical novelty. Perhaps prior to (Kitcher 1983) or

(Lakatos 1976), an account of the practice and methods of mathematics was not on the

agenda of philosophy of mathematics, and only in the last decade or so has the topic been

pursued with vigor and as something that portends important lessons for philosophy.



Thus it would not be entirely insensible for naturalists to seek to account for the methods of

mathematics. And if Maddy’s analysis of mathematical method is correct, then accounting

for the methods of mathematics involves explaining how the facts of mathematical depth

constrain or influence mathematical work. This is all to say that the method-affirming

component of Maddy’s entreaty to understand mathematics on its own terms is not an

unreasonable component of her (or any other) naturalism. Given that certain strands of

modal nominalism—at least Hellman’s Modal Structuralism—are method-affirming in

many senses, and given that modal nominalism certainly does not preclude taking up

the study of the methods of mathematics, it stands to reason that modal nominalism is

compatible with a somewhat stronger version of naturalism that embraces the method-

affirming component of Maddy’s entreaty, but stops short of embracing the method-

contained component.

There is no need to run together, as Maddy appears to do, an account of the methods

of mathematics and an account of the subject-matter of mathematics. That is not to say

that these two items are unrelated—presumably the methods of mathematics help to

illuminate its subject-matter, and its subject-matter constrains its methods. But it would be

implausible to insist that a complete and non-amendable account of the subject-matter of

mathematics could be had solely from a characterization of the methods of mathematics,

just as it would be implausible to insist that a complete and non-amendable account of the

methods of mathematics could be had solely from a characterization of the subject-matter

of mathematics. To play on a famous Lakatos quote, the metaphysics of mathematics,

lacking the guidance of methodology, would become blind, but it would also become blind

without the guidance of philosophy.

All this is to say that although modal nominalism may be inconsistent with Second

Philosophy, nevertheless the modal nominalist should not be bothered by this inconsis-

tency. Second Philosophy appears to be an implausibly strong form of naturalism, and

moreover the naturalistic resources available to the Second Philosopher do not provide any



compelling reasons for supposing that modal nominalism runs into any kind of conflict

with mathematics itself. Given that the only decisive objection the Second Philosopher

can sustain against modal nominalism relies on the strongest and least plausible aspect of

Second Philosophy of mathematics—its excision of extramathematical methods—it would

appear that all weaker forms of naturalism are incapable of producing decisive objections

to modal nominalism. Thus, modal nominalism is compatible with naturalisms that do

not require that mathematics be understood solely using the methods of mathematics.


286

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ABSTRACT

NOMINALISM IN MATHEMATICS: MODALITY AND NATURALISM

by

JAMES S.J. SCHWARTZ

December 2013

Advisor: Dr. Susan Vineberg

Major: Philosophy

Degree: Doctor of Philosophy

I defend modal nominalism in philosophy of mathematics—under which quantification

over mathematical ontology is replaced with various modal assertions—against two

sources of resistance: that modal nominalists face difficulties justifying the modal assertions

that figure in their theories, and that modal nominalism is incompatible with mathematical

naturalism.

Shapiro argues that modal nominalists invoke primitive modal concepts and that they

are thereby unable to justify the various modal assertions that figure in their theories.

The platonist, meanwhile, can appeal to the set-theoretic reduction of modality, and so

can justify assertions about what is logically possible through an appeal to what exists

in the set-theoretic hierarchy. In chapter one, I illustrate the modal involvement of the

major modal nominalist views (Chihara’s Constructibility Theory, Field’s fictionalism, and

Hellman’s Modal Structuralism). Chapter two provides an analysis of Shapiro’s criticism,

and a partial response to it. A response is provided in full in chapter three, in which I

argue that reducing modality does not provide a means for justifying modal assertions,

vitiating the accusation that modal nominalists are particularly burdened by their inability

to justify modal assertions.

Chapter four discusses Burgess’s naturalistic objection that nominalism is unscientific.

I argue that Burgess’s naturalism is inadequately resourced to expose nominalism (modal

or otherwise) as unscientific in a way that would compel a naturalist to reject nominalism.

298

I also argue that Burgess’s favored moderate platonism is also guilty of being unscientific.

Chapter five discusses some objections derived from Maddy’s naturalism, one according

to which modal nominalism fails to affirm or support mathematical method, and a second

according to which modal nominalism fails to be contained or accommodated by math-

ematical method. Though both objections serve as evidence that modal nominalism is

incompatible with Maddy’s naturalism, I argue that Maddy’s naturalism is implausibly

strong and that modal nominalism is compatible with forms of naturalism that relax the

stronger of Maddy’s naturalistic principles.

299

AUTOBIOGRAPHICAL STATEMENT

Schwartz was born and raised in Mid-Michigan and attended Michigan State University for

his undergraduate studies, where he received a B.A. in philosophy in 2007 with cognates

in mathematics and statistics. He completed his graduate work at Wayne State University

where in 2010 he received his M.A. in philosophy and where in 2013 he received his

Ph.D. in philosophy with a minor in mathematics. Schwartz’s primary areas of research

are philosophy of mathematics and modal metaphysics; other areas of interest include

metaphysics and environmental ethics, in particular in the application of environmental

thought to space policy. His publications have appeared in the journals Environmental

Ethics and Ethics and the Environment.

Dissertation - Nominalism in Mathematics: Modality and Naturalism

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